\
SEMIGROUP EXTENSIONS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN
OP WOENSDAG 24 APRIL 1968 TE 14 UUR DOOR
LEO ANTONIUS MARIE VERBEEK
elektrotechnisch ingenieurgeboren te Bergh
UITGEVERIJ WALTMAN - DELFT
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. F. LOONSTRA.
C O N T E N T S
CHAPTER O Preliminary notions
0.1 Semigroups 7 0.2 Mappings and relations 9
0.3 Partially ordered sets and lattices 11 CHAPTER 1 Introduction
1.1 Ideal extensions of semigroups 13 1.2 Schreier extensions of monoids 14
1.3 y4-congruences 17 CHAPTER 2 General theory of semigroup extensions
2.1 Remarks on ideal and Schreier extensions and examples . . . . 19
2.2 Semigroup extensions in general 22 2.3 The set of extensions of ^ by S 28 CHAPTER 3 Union extensions and compositions of semigroups
3.1 Union extensions of semigroups 32 3.2 Compositions of semigroups 35 CHAPTER 4 Existence and construction of union extensions
4.1 Introductory remarks 39 4.2 Union extensions by 1-semigroups 41
4.2.1 The composition S~ = V~ 41 4.2.2 The composition S~ = U~ 42 4.2.3 The composition 5 " = W~ 45 4.2.4 The composition S~ = W;' 49 4.3 Union extensions by 2-semigroups 49
4.3.1 The composition 5 " = f/" u F~ 50 4.3.2 The composition 5 ~ = U~ \j X 52 APPENDIX A Derivation of possible compositions 57
APPENDIX B Examples of compositions 68
References 71 Summary 73 Samenvatting 75 Curriculum vitae 77
CHAPTER O
P R E L I M I N A R Y N O T I O N S
In this chapter the elementary notions and basic properties used in this dissertation are described summarily. The intention of this preliminary chapter is to avoid as much as possible referring the reader to the appropriate literature for checking definitions and notational conventions. In Section 0.1 the notions concerning semi-groups are given. Section 0.2 contains a description of mappings, equivalence rela-tions, homomorphisms and congruences. Section 0.3 gives the notions and properties concerning partially ordered sets and lattices that will be used.
0.1 Semigroups
The description of the notions given in this section is based on CLIFFORD and PRESTON [1961].
A partial groupoid S (o) is a non-empty set S together with a partial binary operation (o) such that if Jo? is defined for s and t in S, then Sct is an element of S. We shall say that S is a partial groupoid with respect to (o). The underlying set S of a partial groupoid 5 (o) is called its carrier. A groupoid S (o) is a partial groupoid in which the binary operation (o) is defined for all 5' and / in S. The explicit indication of the opera-tion (o) will be omitted unless this might lead to confusion. Hence we usually write St instead of Sot, and we shall use the symbol S to denote the groupoid and also to denote its carrier. We shall call st the product of the elements s and tofa groupoid S. If A and B are two non-empty subsets of a groupoid S then the product AS is defined as AB = {ab: a e A, be B}. \ï se S and ^4 is a non-empty subset of S, then the product sA is defined &% sA = {sa: ae A] and analogously. As = {as: a e A). Hence the word "product" has several interpretations.
A subgroupoid Tofa groupoid 5 is a non-empty subset of S such that the product of any two elements of T is contained in T, that is, such that TT ^ T. A semigroup S is a groupoid the binary operation of which is associative, that is, for all r, s and t in S holds (rs)t = r(st). A subgroupoid 7' of a semigroup S is a semigroup and is called subsemigroup of S, T \s a proper subsemigroup of S if T is properly contained in S, that is, if r c 5 and T ^ S. The cartesian product AxB of two non-empty sets A and B is the set of ordered pairs (a, b) with a in A and b in B. The direct product S ® T of two semigroups S and T is the semigroup with the set 5 X T as carrier in which the product is defined as {a,b){a',b') = {aa',bb') for all elements {a,b) and {a',b') in SxT. An ideal A of a semigroup 5 is a non-empty subset of S such that AS c s and SA ^ 5. Each ideal of a semigroup 5 is a subsemigroup of S.
An element ^ of a semigroup 5 is a left zero element of S if qs = q for each s in S. An element o of a semigroup 5 is a zero element of S if as = so = o for each j in S. An element of a semigroup 5 is a zero element if and only if it is a right and a left zero element of 5. A semigroup contains at most one zero element. If a semigroup 5 contains a zero element then it is the only left zero element and the only right zero element of S. An element 5 of a semigroup 5 containing a zero element o is a proper zero divisor if j # o and there is an element t ^ o \n S such that st = o and/or ts =^ o. S contains no proper zero divisor if and only if S\o is a proper subsemigroup of 5. An element j of a semigroup S is said to be idempotent if ss = s. An element e of a semigroup S is an identity element of S if es = se = s for each ^ in 5. A semigroup contains at most one identity element. A monoid is a semigroup containing an identity element. A submonoid of a monoid 5 is a subsemigroup of S containing the identity element of S.
Let 5 be a semigroup. Extend the binary operation on S to the set 5 u {1}, where 1 is an element not contained in S, by defining 11 = 1 and 15' = si = j for each s in 5. Clearly, 5 u {1} is a monoid with respect to this extended operation and 1 is its identity element. The described manipulation is called the adjunction of 1 as identity element to the semigroup S. If S is a semigroup, then 5 ' denotes the monoid obtained by the adjunction of an identity element to 5 in case S does not contain an identity element, and 5 ' = 5 in case 5 is a monoid.
Let 5 be a semigroup. Extend the binary operation on 5 to the set S u {o}, where o is an element not contained in S, by defining oo = o and os = so = o for all s in S. Evidently, S u {o} is a semigroup with respect to this extended operation and o is its zero element. This manipulation is called the adjunction of o as zero element to the semigroup S.
Let ^ be a non-empty set. Define a binary operation (o) on A by aob = b for ail a and è in ^4. ^ is a semigroup with respect to (o) and A (o) is called the right zero semigroup on the set A. Clearly, the right zero semigroup on A is unique. Each ele-ment of a right zero semigroup is a right zero eleele-ment. Similarly, if we define on a set A a binary operation {o) by aob = a for all a and b in A, then A (o) is a semigroup called the left zero semigroup on A. The left zero semigroup on a set A is unique. Each element of a left zero semigroup is a left zero element.
A semigroup S is commutative if st = ts for all s and / in S. ^ semigroup 5 is cancellative if xs = xt or sx = tx entails s=tfoT all s, t and x in S. A semigroup S is weakly reductive if, for s and t in 5, xs = xt and sx — tx for all x in S implies s = t. Two elements s and r of a semigroup S are inverses of each other if sts = s and tst = t. An inverse semigroup is a semigroup each element of which has a unique inverse.
A semigroup S is simple if S contains no proper ideal. If 5 contains a zero element o then S is o-simple if 5 5 / o and o is the only proper ideal of 5. Let E be the set of idempotent elements of a semigroup 5. Let e,fe E, define e < ƒ if and only if ef=fe =
= e. Then < is a partial ordering of E. If 5 contains a zero element o then o -^ e for every e in E. An idempotent element ƒ of 5 is primitive if f ^ o and e < ƒ implies e = o or e —f. A completely simple semigroup 5 is a simple semigroup containing a primitive idempotent element. A completely o-simple semigroup 5 is a semigroup containing the zero element o and such that 5 is o-simple and contains a primitive idempotent element. A Brandt semigroup is a semigroup 5 with zero element o such that 5 is a completely o-simple inverse semigroup.
0.2 Mappings and relations
The concepts and properties given in this section can be found for instance in CLIF-FORD and PRESTON [1961].
A mapping a: A -> fl of a set /4 into a set Bis a correspondence which associates with each element a in A one element aa in B. If no confusion can arise we often write a instead of a: ^4 -> B. The element aa is called the image of a under the mapping a. If b e 5 then the pre-image of b is the subset ba~^ = {ae A: aa = b} of A. If ba~' ^ 0 for each b in B, then a is a mapping of A onto B. A mapping a is a one-to-one mapping of A onto B if each element of B is the image of precisely one-to-one element of A, then also a~^: B ^ A is a one-to-one mapping of B onto A.
A (binary) relation Q ona set A is a subset of the cartesian product Ax A of A with itself. Let Q and a be two relations on A. The composition QOC of Q and a is the relation on A defined as {a,b)e QOU if and only if there exists an element cin A such that {a,c) e Q and {c,b) e a. Clearly, the composition of relations on A is associative, that is, (eoö-)oT = ÖO((TOT). The identity relation i is defined as i = {(fl,a): a e A\. The converse Q~^ of a relation g on ^ is defined by (a,b)e Q~^ if and only ifib,a) e Q. Clearly, (Q~^)~^ = Q and (g o a ) ~ ' = a~' o ^ ~'. A relation g is contained in a relation (T if e is a subset of a, that is, Q ^ a. \n other words, Q ^ a if and only if {a,b) e g entails {a,b) e a. A relation g on A is reflexive if (a,a) £ g for all a in A, that is, i ^ g; g is symmetric if from (a,b)eg it follows that {b,a)eg, that is, g ^ g~^ (hence g = e ~ ' ) ; g is transitive if from ia,b) e g and {b,c) e g it follows that (a,c) e g, that is, g o g ^ g.
A reflexive, symmetric and transitive relation on A is called an equivalence relation. Clearly, the identity relation t on ^ is an equivalence relation, moreover, i is con-tained in every equivalence relation on A. Let g be an equivalence relation on a set A. If {a,b) e g we say a and b arc equivalent modulo g, or also, a and b are g-equivalent. The set of all elements in A which are g-equivalent to a is denoted by a", so that a* = {h e A:(a,b) e g}; a- is called the equivalence class of A modulo g containing a or the g-class containing a. Clearly, a e a" for every a in A, and if a' n b^ ¥= 0 then cfi = b". Hence the set of all gi-classes is a partition of A, that is, they are mutually disjoint and their union is A. This partition is denoted by Ajg and is called theyac^or set of A modulo g. The mapping a -* a" is called the natural mapping of A onto A/g
and is denoted by g„^,. Notice that a" = ag„^i for each a in A. Conversely, a partition ^ of a set A determines an equivalence relation g on ^ such that ^ — Ajg, namely {a, b)e gif and only if a and b are contained in the same element of ^.
Let a be a mapping of a set A into a set B. Then a can be regarded as a relation on A \j B. The relation a o a ~ ' is contained in Ax A, and is defined by (a,a') e aoa~', with a and a' in A, if and only if aa = a'a. Clearly, aoa"^ is an equivalence relation on A. Hence a induces in an obvious way a one-to-one mapping of A/aoa~^ onto Aa c B. The relation « o a " ' on A is called the equivalence relation on A naturally induced by the mapping a.
Let g be an arbitrary relation on a set A. The transitive closure gT of g is defined as
gT = \J g" = g ^ (gog) (J (gogog) u ... n= I
Clearly gT is the smallest (with respect to inclusion) transitive relation containing g. If go is any relation on A, the relation g^ = g^ <j QQ ^ u i is the smallest reflexive and symmetric relation on A containing gg. The transitive closure g = giT of gi is the smallest equivalence relation on A containing go; g is called the equivalence relation on A generated by gg.
The intersection of any set of equivalence relations on a set A is an equivalence relation on A, but this is not true for the union of even two equivalence relations. The Join g V a of two equivalence relations g and ff on ^ is the equivalence relation
generated by g u cr. Since g u CT is reflexive and symmetric, g w a is the transitive closure of g u CT.
Let A and B he partial groupoids. A mapping a: A ->^ B of A into or onto fi is a partial homomorphism of A into or onto B, respectively, if for any two elements a and bin A such that ab is defined the product {aa){ba) is also defined and {aa){ba) = {ab)a. A mapping a: A -* B of A onto B is an isomorphism of A onto fi if a is a one-to-one homomorphism of A onto B and a " ' is a homomorphism of Ö onto A. In this case A and B are said to be isomorphic, denoted hy A = B.
Let 5 and T be semigroups and let a: 5 -> J be a homomorphism of 5 into T, so that {aa){ba) = {ab)a for all a and b in 5. The image 5a of 5 under the homomorphism a is a subsemigroup of T. If 5 is a monoid with identity element e, then the sub-semigroup 5a of r i s a monoid with identity element ea. An isomorphism a ofS into T is a one-to-one homomorphism of 5into T. If 5, Tand [/are semigroups and a: 5 -> T and p: T -* U are homomorphisms of 5 into T and of T into U, respectively, then aP: S ^ [/ is a homomorphism of 5 into U. An isomorphism of a semigroup 5 onto itself is called an automorphism of 5.
A congruence g on a semigroup 5 is an equivalence relation on 5 such that from (s,t)eg and {u,v)eg it follows that (su,tv)eg. This definition is equivalent t o : a congruence g on a semigroup 5 is an equivalence relation on 5 such that (s, t)e g entails {xsy,xty) e g for all x and y in 5 ' . An equivalence class of 5 modulo a
con-gruence g is called a concon-gruence class of 5 modulo g, or g-class of 5. Let j and / be two arbitrary elements of 5 so that s" and t" are two elements of S/g. It is easily verified that the product (5^)(?^) is contained in the g-class (st)'. Now define on S/g a binary operation as (sg){tg) = (st)", then S/g is a semigroup. This semigroup is called the factor semigroup ofS modulo g and is denoted by S/g. Let g„^^ again denote the natural mapping of 5 onto S/g, then s' = sg„^i for all j in 5 so that (jgnaOC^ffnat) =
= (^Oönaf Hence g„3, is a homomorphism, it is called the natural homomorphism of 5 onto S/g. Let 5 be a monoid with identity element e. If g is a congruence on 5 then S/g = 5g„a, is called the factor monoid of 5 modulo g. The identity element of Sg„,i is e" = eg„^^.
Let 5 and Tbe two semigroups and let a: 5 -> T be a homomorphism of 5 into T. It is easily verified that the equivalence relation a o a " ' on 5 is a congruence on 5, a o a ~ ' is called the congruence on S naturally induced by a. Moreover, the factor semigroup 5/aoa~' is isomorphic with the image 5a ^ T of 5 under the homo-morphism a.
Let g and cr be two congruences on a semigroup 5 such that g ^ a. Consider the mapping g~Ja„^^ of S/g = Sg„^^ into S/a = 5CT„3,. For each element sg^^, of 5e„^i, where s e 5, holds sg„^^g^.^l(T„^^ = iCnat- Clearly, ffMt'o'nat is a homomorphism of 5g„a, onto 5ff„a,.
Let I be an ideal of a semigroup 5. Define on 5 the relation g as (s, t) e g, for s and ? in 5, if either s = t or else s and ? are both in /, that is, g = {(s, t)e SxS: s = t or s,t e ƒ}. Clearly, g is a congruence on 5. It is called the Rees congruence on 5 modulo I. The g-classes of 5 are I and every one-element set {s} with j in 5\/. We write 5/7 instead of S/g, and call S/I the Rees factor semigroup of S modulo I.
0.3 Partially ordered sets and lattices
The presentation of the notions and properties in this section is based on CLIFFORD and PRESTON [1961] and on HERMES [1955].
A relation g on a set A is anti-symmetric if from {a,b) e g and {b,a) eg it follows that a = b, that is, if g n g~^ c (^ where i is the identity relation on A.
A relation ^ on a set A is a partial ordering of A, or A is partially ordered by ^ , if < is reflexive, anti-symmetric and transitive. Instead of (a, è) e < we write a < 6 or b ^ a. An element a of A is an upper bound of a subset B of A if b ^ a for each b in B. An upper bound a of a subset B of A, such that a < c for each upper bound c of B, is called the supremum or Join of B, denoted by sup B. \f B has a supremum it is unique. Similarly, a lower bound of a subset 5 of .4 is an element a in A such that a^bfor each element b in B. A lower bound a of B, such that Ö > c for each lower bound c of B, is called the infimum or wee? of 5, denoted by inf B. If 5 has an infimum it is unique.
A has a meet, denoted by a A b. Then each finite subset of A also has a meet. An upper semilattice ^4 is a partially ordered set A such that every two-element subset {a,b} of A has a join, denoted by a v b. Then each finite subset of A also has a join.
A lattice is a partially ordered set which is both a lower and an upper semilattice. The partial ordering < and the two binary operations A and v of a lattice A are related as follows: for all a and bin A holds if a ^ b then a A b = a and a v b = b, and if aAb = a or avb = b then a ^ b. A lattice A is complete if every subset of
A has a meet and a join. Then A also contains its meet, inf A, and its join, sup A. A sublattice B of a lattice ^4 is a non-empty subset B of A such that B contains with
each two-element subset {a, b} of A also a A b and a v b. A complete sublattice B of a lattice ^ is a sublattice of A which contains with each subset X of A also inf X and sup X.
Let A and B be two sets partially ordered by < and < , respectively. A mapping
a: A ^ B of A into B is an order-homomorphism if from x < ƒ for x and >> in ^4 it
follows that xa -< ^a. If a: y4 ^ B is a one-to-one order-homomorphism of A onto
B and a " ' : B -* A is an homomorphism of B onto ^ , then a is an order-isomorphism, and /4 and 5 are said to be order-isomorphic.
Let A and B be lattices. A mapping a: A -> B of A into 5 is a lattice-homomorphism if a is such that x a A > ' a = ( x A y)a and xa v j ; a = (x v j ) a for all x and ƒ in A. If a: /4 -> 5 is a lattice-homomorphism then it is also an order-homomorphism. A mapping a: A -> Bof A onto J5 is a lattice-isomorphism of /4 onto B if a is a one-to-one lattice-homomorphism, in this case A and B are said to be lattice-isomorphic. A mapping a of a lattice A onto a lattice fi is a lattice-isomorphism if and only if it is an order-isomorphism.
Let y4 be a lattice and let ^ be the partial ordering of A. A non-empty subset / of ^4 is a A -ideal of A if the following conditions are satisfied:
i. if x,y e I then x A ye I, and ii. if X6 / a n d ae A then x y ae I.
An equivalent definition is: a non-empty subset / of a lattice /4 is a A -ideal of A if
x,y 6 ƒ if and only if x A y e I.
Clearly, a A-ideal of a lattice A is always a sublattice of ^4. I f / i s a A-ideal of the lattice A and x is in I, then every element a ^ x is in /. Let a be an element of the lattice A. The set B =^ {b e A: b "^ a] is a A -ideal of A and B is called the principal
CHAPTER 1
I N T R O D U C T I O N
In this chapter we give, in the Sections 1.1 and 1.2, a short account of the definition and the basic construction of ideal extensions of semigroups and of Schreier exten-sions of monoids based on CLIFFORD [1950] and on RÉDEI [1952], respectively. Also the contents of later publications on these two types of extensions is indicated. In Section 1.3 a particular set of congruences on a semigroup is defined and its properties are described insofar as we need these later on. This section is based on CLIFFORD and PRESTON [1967].
1.1 Ideal extensions of semigroups
Let A and 5 be disjoint semigroups and let 5 contain a zero element o. A semigroup E is an ideal extension of A by S if it contains A as an ideal and if the Rees factor semigroup E/A is isomorphic with 5. Clearly, the carrier of any ideal extension of A by 5 is the union of A and a partial groupoid isomorphic with 5 " = S\o. We follow common usage by considering only the set £ = ^ u 5 " as the carrier of ideal exten-sions of A by 5. Then every ideal extension £(0) of ^4 by 5 is obtained by finding all possible ways in which an associative binary operation (o) can be defined on the set E = A u S~ such that the following conditions hold for all a and b in A and all s and t in S~:
aob = ab, aoseA, SoaeA and
!
= st if st =^ o e A if st = o.
The preceding remarks contain all information concerning ideal extensions of semigroups that we shall use. They are taken from CLIFFORD [1950], and are also given in Section 4.4 of CLIFFORD and PRESTON [1961]. It may be interesting to indicate the main problems concerning ideal extensions that have received attention in the literature. Because of the rather complicated nature of the methods and conditions used in the publications, and since we do not need these in our investigations, we describe those problems only summarily and refrain from stating the results obtained.
One question concerns the determination of all ideal extensions of a given
semi-group ^ by a given semisemi-group 5 containing the zero element o. In CLIFFORD [1941] and [1950] this question is investigated in case A is a completely simple semigroup and 5 is arbitrary, a part of the results is also given in CLIFFORD and PRESTON [1961]. In case ^ is a group and 5 is a completely o-simple semigroup the problem has been studied by MUNN [1955], the results are published in Clifford and PRESTON [1961].
In WARNE [1966b] a solution to the same problem is given in case y4 is a Brandt semi-group and 5 is an arbitrary semisemi-group. Closely related with these investigations are the papers by PETRICH [1966] and by WARNE [1966a] in which necessary and sufficient conditions on A and 5 are derived for all ideal extensions of ^ by 5 to be determined by partial homomorphisms of the partial groupoid 5 ~ = 5\o into the semigroup A. Each of these two publications is concerned with cases where A and 5 satisfy par-ticular conditions.
Another question concerns the existence of ideal extensions. If two disjoint semi-groups A and 5, where 5 contains a zero element, are given, it is not always possible to find an ideal extension of A by 5. Necessary and sufficient conditions on A and/or 5 for the existence of an ideal extension of ^ by 5 are not known in general. How-ever, if A contains an idempotent element or if 5 does not contain a proper divisor of 0, then rules for the construction of an ideal extension of ^ by 5 can be given explicitly (see exercises for Section 4.4 in CLIFFORD and PRESTON, [1961]). HEUER and MILLER [1966] investigate necessary and suflacient conditions for the existence of a cancellative ideal extension of a cancellative semigroup by a group with a zero element adjoined.
Still another problem concerns the equivalence of ideal extensions. Two ideal extensions E and E' of a semigroup A by a semigroup 5 containing a zero element are equivalent if there exists an isomorphism of E onto E' mapping the ideal ^4 of £ onto the ideal A of £", that is, the isomorphism restricted to A is an automorphism. Necessary and sufficient conditions for the equivalence of ideal extensions of ^ by 5 in case ^4 is a monoid and the extensions are determined by partial homomorphisms of 5 ~ into A are given in CLIFFORD and PRESTON [1961]. In case A is a group and 5 is a completely o-simple semigroup these conditions are worked out in more detail by MUNN [1955] and published in CLIFFORD and PRESTON [1961]. In relation herewith we mention the paper by INASARIDZE [1966] in which methods from categorical algebra are used in an investigation of ideal extensions of semigroups.
1.2 Schreier extensions of monoids
Let M be a monoid with identity element e and, possibly, containing a zero element o. Let e be a congruence on M. The congruence class N = e" of M modulo g containing e is called the principal class of the partition M/g or also the principal class of g. Clearly, TV is a submonoid of M, moreover, N is the identity element of the factor monoid M/g. The congruence g is, in general, not determined by its principal class N, although in case M is a group g is determined by N. In order to ensure that g is deter-mined by its principal class the notion of left normality is introduced. For this we need the following. The product AB of two subsets A and B of a monoid M is said to be without repetition if ab = a'b', where a, a' e A and b, b' e B, entails a = a' and b = b'. Notice that if M contains a zero element o, then oB ( = o) is not without
repetition in case B contains more than one element. The partition M/g of a monoid M with identity element e modulo a congruence g with principal class Nis a left normal partition if the g-classes are of the form:
(1) fliTV, ajN, a^N, ..., where a, = e and every product aiN, with a,- / o, is without repetition.
If we consider an element Z>; e OiN we can easily verify that b^N c a.yv, whereas biN = a^N for all 6,- e a^N and all a, Af c M if and only if A'^ is a group. From this it follows that «2' ^3> • • • ars> i" general, not arbitrary elements of the congruence classes « j ^ - '^sM ••• of (U- Now one easily verifies the following
Property. Let g be a congruence on a monoid M and let A' be the principal class of g. If M/g is a left normal partition, then it is the only left normal partition with principal class A^, that is, then it is uniquely determined by A^.
In proving this property one does not need the fact that the products a^N for a,- # o are without repetition. The restriction of the concept of left normality to congruence classes without repetition is introduced to simplify the theory of Schreier extensions which is based on it. A submonoid TV of a monoid M is said to be a left normal submonoid of M if N is the principal class of a left normal partition of M modulo a congruence on M. If A'^ is a left normal submonoid of M, then (1) is called the partition of M modulo N and the associated factor monoid is denoted by M/N.
Let A and 5 be disjoint monoids with identity elements e^ and eg, respectively. A monoid £ is a Schreier extension of A by S if E contains a left normal submonoid A ^ A and E/A ^ S. If E contains a zero element o^, then 5 contains also a zero element Og. We restrict the discussion to the case that if 5 contains a zero element Og then E contains also a zero element o^; later, see the Remark following the Theorem, we shall indicate how Schreier extensions without zero element may be obtained. Whether A contains a zero element or not is irrelevant.
Clearly, there exists always a Schreier extension of a monoid ^4 by a monoid 5, to wit the direct product of A and 5. In order to find all Schreier extensions of A by S the following construction is used. Consider all (ordered) pairs (s,a) with j in 5 and a in A, and identify all pairs {os,a). Here and in the sequel the statements concerning Os are to be neglected in case 5 contains no zero element. On the set of these pairs (s,a) we define a (not necessarily associative) binary operation (o) in the following way: (2) {s,a)o{t,b) = (st,f(s,t)g{a,t)b) for all s,teS and all a,b e A, where (3) ƒ is a mapping of the cartesian product SxS into A,
g is a mapping of the cartesian product AxS into A, and ƒ and g satisfy the conditions
(4) f{s,es) = e^, f(es,s) = e^, g(a,es) = a, ^(e^,j) = e^.
Schreier product of S and A. Since all pairs (os,a) are one and the same zero element of So A it follows from (2) that the image of the mappings ƒ and g for st = Oj and for / = Os, respectively, is irrelevant so that they can be assigned arbitrarily. The element {es,e^ is the identity element of So A due to condition (4). From (2) and (4) it follows that
{s,e^)o{t,e^) = ist,f{s,t)) and (gj,a)o(r,e^) = {t,g(a,t)).
This result, taken together with (2), shows that the mappings of (3) and the Schreier product 5 o ^ completely determine each other. The relation between the Schreier products and the Schreier extensions is given in the following
THEOREM. (RÉDEI [1952]) Let A and 5 be disjoint monoids with identity elements e^ and e^, respectively.
(i) A Schreier product E= SoA is a monoid if and only if for all a,beA and a\\ s,t,ue S the following conditions hold:
(5) g(ab,s) = g(a,s)gib,s) for s # Og,
(6) g{a,st)f(s,t)=fis,t)g(gia,s),t) for st ¥= Og, (7) f(s,tu)f(t,u)=f(st,u)g(f{s,t),u) for stu ¥^ Og.
(ii) These monoids are, up to isomorphism, all Schreier extensions of A by 5. In particular, the elements ies,a) with a in A form a left normal submonoid A of E such that
(8) A ^ A under the mapping (es,a) -y a, and E/A ^ 5 under the mapping (s, e^)A -• s.
(iii) A Schreier extension E = SoA is a group if and only if both 5 and A are groups.
Remark. In case 5 contains a zero element Og we have considered only Schreier extensions E containing the zero element ios,a) with a in .4. If one wants to deter-mine the Schreier extensions without zero element the construction has to be modified as follows: All pairs (s,a) are considered as distinct elements of SoA; the image of the functions ƒ and g are, for st = Og and t = Og, respectively, not arbitrary; the condi-tions (5), (6) and (7) have to be valid for all s, t and M in 5.
The exposition given above, based on § 3 and § 4 of RÉDEI [1952], is closely related to the Schreier theory of group extensions. In Rédei's paper many further results concerning Schreier extensions are worked out and we shall now indicate these briefly.
Several corollaries of the Theorem in case the mappings ƒ or g of (3) are of a par-ticular simple form are given.
From a given Schreier extension £ of a monoid ^ by a monoid 5 determined by a pair of mappings ƒ and g, Rédei derives a collection of pairs of mappings/' and g',
each pair determining a Schreier extension £ ' of A by S such that £ ' is isomorphic with £.
Let £ and £ ' be two Schreier extensions of /I by 5, let a: ^4 -> £ and a': ^4 -^ £ ' be the two isomorphisms of A onto the left normal submonoids ^4 of £ and A' of £ ' , respectively, and let P and P' he the two homomorphisms of £ and £ ' , respectively, on 5. £ is equivalent to £ ' if there exists an isomorphism y of £ onto £ ' such that ay = a' and P = yP'. The equivalence of Schreier extensions is investigated by RÉDEI in terms of the mappings ƒ and g of (3).
Finally, the Schreier extension theory of semirings with zero element is worked out analogously to the extension theory of monoids. A semiring 5 is a set with two binary operations, ( + ) and ( x ) , where 5 is a commutative and cancellative monoid with respect to ( + ) and a semigroup with respect to ( x ) and the usual distributive law of ( + ) and ( x ) holds.
We shall now mention the problems considered in later pubhcations concerning Schreier extensions of monoids. WIEGANDT [1958] investigates the characterization of a cancellative monoid A which forms a direct factor of every cancellative Schreier extension of .4. The same problem for commutative and cancellative monoids, called semimodules, is investigated by HANCOCK [I960]. In HANCOCK [1964] commutative Schreier extensions of a group by a commutative monoid and several related topics are studied. INASARIDZE [1965] studies Schreier extensions of semimodules by semi-modules using categorical algebra.
1.3 A-congruences
The contents of this section is based on Section 10.1 of CLIFFORD and PRESTON [1967] where references to the original publications are given.
Let 5 be a semigroup and let C be the set of all congruences on 5. Recall that a congruence g is a subset of a congruence a, g ^ a, if (s, t)e g entails {s, t)ea where i and / are elements of 5. It is evident that the inclusion relation ^ is a partial ordering of C. If g and a are in C, then clearly also their intersection g r\ a isin C. If g; e C, where i e I and / is an index set, then also n {g,: iel) eC, that is, for every subset C' of C also inf C' e C. Furthermore, SxSe C. Hence , if we define the operation
V on C in the usual way as
V {gj: iel) = n {g: geC, g 3 g; for all iel],
then C becomes a complete lattice with respect to n and v . If g and a are in C it is now evident that the join g v a of g and a is the transitive closure of the union of g and a, that is, g v a = {g yj a) T.
Let ^ be a non-empty subset of a semigroup 5. If there exists a congruence g on 5 such that ^ is a congruence class of 5 modulo g, then A is called an admissible subset of 5, and g is said to be an A-congruence. If A is an admissible subset of 5 there may
be more than one ^-congruence on 5. We denote the set of all ^-congruences on 5 by C(A). It is easily verified that the infimum of any subset of C(A) is an .4-congruence. Also, if g and a are in C(A), then their join g v o- is in C(A). Hence, C(A) is a com-plete sublattice of the lattice C of all congruences on 5.
In order to find the infimum and the supremum of C{A) and at the same time criteria for the admissibility of a subset A of 5 we proceed as follows. Let .4 be a non-empty subset of a semigroup 5. Consider the relation
^ = {(a,b)eSxS: a = è or else a e x ^ j and é e x / I j for some x , > ' e 5 ' } and its transitive closure n = £,T.
It is easily verified that ^i is a congruence on 5 and that A is contained in one /j-class. Moreover, /i is the least congruence on 5 such that A is contained in a single congruence class. Now consider another relation on 5, namely
V = {{a,b)eSxS: xayeA if and only if xby e A for all x, j e 5 ' } .
One easily verifies that v is a congruence on 5 and that A is the union of I'-classes. Moreover, v is the greatest congruence on 5 such that A is the union of congruence classes. Let us now suppose that A is an admissible subset of 5 and that g is a con-gruence on 5. Evidently, ^4 is a g-class if and only if A is both a union of g-classes and A is contained in a g-class. Hence it is clear from the above remarks that g is an ^-congruence if and only if n '^ g ^ v. Moreover, A is admissible if and only if ii ^ V. An equivalent criterion is: A is admissible if and only if xAy n A ^ 0 entails xAy E A, where x, jFe5'. It is now evident that in case A is admissible, that is, C(A) is non-empty, then the infimum and supremum of C(y4) are /i and v, respec-tively.
CHAPTER 2
G E N E R A L T H E O R Y O F S E M I G R O U P E X T E N S I O N S
In this chapter we develop a general theory of semigroup extensions. Section 2.1 contains some observations and examples concerning ideal and Schreier extensions and then two additional examples. These motivate a proposal for a general definition of semigroup extensions, the implications of which are worked out in Section 2.2. In Section 2.3 the set of all semigroup extensions of one semigroup by another is studied, a partial ordering related with this set is introduced and investigated in some detail, and an example illustrating most of the derived properties is given.
2.1 Remarks on ideal and Schreier extensions and examples
First we consider ideal extensions. Let A and 5 be two disjoint semigroups and let 5 contain the zero element o. By 5 " we denote 5\o. Let £ be an ideal extension of A by 5. Then the carrier £ = y4 u 5 ~ and the Rees congruence 6 = {{a,b) e ExE: a = b or a, b e A} on E modulo A is an /I-congruence. Since the ö-classes are A and every one-element subset of E\A it is evident that 0 is the least (with respect to congruence inclusion) ^-congruence on £. Hence there may be /<-congruences Ö' on £ such that Ö C Ö'.
An example of this situation is given by the Cayley tables of Figure 1. Table A gives a group A = {e^,a} and Table 5 gives a monoid 5 = {es,s,o} with zero element o. Any ideal extension E of A by S consists of four elements, £ = ^ u 5 ~ = {eg,s,e^,a}. An associative operation on £ is given in Table E of Figure 1. The Rees congruence 0 on £ modulo A has as ö-classes the sets {cg}, {s} and {e^,a} and E/6 is isomorphic with 5. Now consider the .4-congruence 6' on £ with ö'-classes {es,s} and {e^,a}. Clearly, E/6' is isomorphic with the semigroup T = {t,u} given in Table Tof Figure 1. Since 0 ^ 6' it is evident that T{^ E/9') is also a homomorphic image of 5 ( ^ E/6). Hence we might consider £ as being constructed from A and T in the sense that £ contains A as ideal and there exists an y4-congruence Ö' on £ such that E/0' = T.
SA a «A ^A a a a eA Table A es s 0 ^s es s 0 s s s 0 0 0 0 0 es s eA a es es s eA a s s s eA a eA eA eA eA a a a a a eA t u t t u u u u Table T Table 5 Table £
Now we consider Schreier extensions. Let A and 5 be two disjoint monoids with identity elements e^ and eg, respectively. Let £ be a Schreier extension without zero element of A by 5. According to Rédei's Theorem and Remark given in Section 1.2 the monoid £ is isomorphic with a monoid £ ' with carrier E' = SxA. Moreover, £ ' contains the left normal submonoid A = {(eg,a): aeA}, where A = A, and for each element {s,e^) in £ ' holds (s,e^)A = {(s,a): aeA}. Furthermore, the congruence g on £ ' determined by A has as g-classes the subsets {{s,a): aeA} of £ ' . In case 5 contains a zero elemento o the g-class {{o,a): aeA} is clearly an ideal of £ ' and one obtains a Schreier extension F' with zero element z of A by S by identifying all ele-ments of this ideal so that z = {{o,a): aeA}. (This means that £ ' is the Rees factor monoid of £ ' modulo this ideal of £'). Hence a Schreier extension £with zero element of Y4 by 5 is isomorphic with a monoid F' with zero element z, where the carrier of F' is { 5 - x ^ } u {z} and 5 " = S\o.
Suppose £ is a Schreier extension of A by 5. Whether £ contains a zero element or not, it is evident from the previous remarks that the congruence 6 on £, determined by the left normal submonoid N = A of £, is the least A^-congruence on £. Hence there may be A^-congruences 6' on £ such that 6^0'.
In Figure 2 we give an example of this situation by means of Cayley tables. Table A gives the group A = {e^, a} and Table 5 gives the monoid 5 = {eg,s,o} with zero element o. Notice that A and 5 are the same as in the example of Figure 1. In Table £ a Schreier extension £ = {1,2,3,4,5,6,} without zero of ^4 by 5 is given. The
iso-1 -^ (es^e^) 2 -^ (eg,a) 3 -* is,e^) 4 -^ (s,a) 5 - io,e^) 6 -+ (o,a) eA a eA «A a a a eA Table A es s 0 es es s 0 s s s 0 0 0 0 0 Table 5 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 1 4 3 6 5 3 3 3 3 3 6 6 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 Mapping £ ->^ £ ' Table £ 1 2 3 4 J 1 1 2 3 4 J' 2 2 1 4 3 >' 3 3 3 3 3 y 4 4 4 4 4 y y y y y y y 1 -^ {eg,e;) {es,a) i.s,e^) {s,a) z 2 3 4 y t u t t u u u u Table T Mapping F ^ F' Table F
morphism mapping £ onto E', with carrier SxA, is given in Mapping E ^ E' of Figure 2. Notice that £ is not the simplest Schreier extension of ^4 by 5 for that would be their direct product S ® A. A Schreier extension F = {1,2,3,4,>'} with zero element ƒ of ^4 by 5 is given in Table £. The isomorphism of F onto £', with carrier {S' XA} u {z} where z = {(o,e^), (o,a)}, is given in Mapping F ^ F' of Figure 2. The Schreier extension £ of ^ by 5 contains the left normal submonoid A'^ = {1,2} and N ^ A. The congruence Ö on £ determined by A^ has ö-classes {1,2,} {3,4} and {5,6} and clearly £/ö = 5. Consider now the A^-congruence 6' on £ with ö'-classes {1,2} and {3,4,5,6}. Clearly £/ö' is isomorphic with the monoid T= {t,u} given in Table T of Figure 2. Since 6 ^ 6' it is evident that T is also a homomorphic image of 5. One might consider £ as being constructed from A and T in the sense that £ contains a left normal submonoid A^, where N = A and there exists an A'^-congruence 9' on £ such that £/ö' = T. For the Schreier extension F with zero element of ^ by 5 one can make observations analogous to those just made concerning £.
The examples given in Figures 1 and 2 show that an ideal or a Schreier extension E of A by S may sometimes be regarded as constructed from A and a semigroup T, where T is a homomorphic image of 5, in the sense that £ contains a subsemigroup A = A and there is an .4-congruence 6' on £ such that £/0' = T. There are also other cases in which a semigroup £ can be considered as being constructed in this same sense from two semigroups A and 5, but without £ being an ideal or a Schreier exten-sion of ^ by a semigroup which has 5 as a homomorphic image. Two such cases are demonstrated by the Cayley tables of Figure 3. Table A gives a group A = {e^,a}, which is the same as in Figures 1 and 2, and Table 5 gives a monoid 5 = {es,s,t,u} wihout zero element. The monoid £ = {b,c,d,f,g,h} given in Table £ contains the subgroup A = {b,c} and A = A. The least i4-congruence cr on £ clearly has as ff-classes the sets {b,c}, {d,f}, {g} and {h}. Moreover, E/a ^ 5. Since 5 contains no zero element, £ is not an ideal extension of ^ by 5, nor of ^ by some semigroup which has 5 as homomorphic image. £ is not a Schreier extension of A by 5, nor of A by some semigroup which has 5 as homomorphic image, because £ has too few elements. Now consider the monoid £ = {1,2,3,4,5,6,7,8} given by Table F of Figure 3. £ contains the subgroup G = {1,2} and G = A. The least G-congruence p on £ has as ^i-classes the sets {1,2}, {3,4}, {5}, {6}, {7} and {8}, but £/^ is not isomorphic with 5. However, the subsets {1,2}, {3,4}, {5,6} and {7,8} of £ determine a G-congruence 6 on F with just these subsets as Ö-classes. Moreover, F/6 = 5. Again one sees that, since 5 contains no zero element, F is not an ideal extension of A by 5, nor of A by some semigroup which has 5 as a homomorphic image. Also, £ is not a Schreier extension of ^4 by 5 because 6 is not the least G-congruence on F, that is, Ö # ^. Finally, £ is not a Schreier extension of A by some semigroup which has 5 as homo-morphic image because F has too few elements.
In Section 2.2 we shall need the following observation. Besides G also G' = {5,6} is a subgroup of the semigroup £such that G' ^ A and Ö is a G'-congruence.
More-over, G" = {5,7} is another subgroup of F such that G" ^ A, but 6 is not a G"-congruence. eA a eA eA a a a eA Table A es s t u eg s eg s s t t u u t t t u t u u u t u t Tables 5 b c d
f
g h b b c df
g h c c bf
d g h d ƒ d ƒ d ƒ g g g g h h g g g g g h h g h h h h g g h g Table £ 1 2 3 4 5 6 7 8 1 1 2 3 4 5 6 7 8 2 2 1 4 3 5 6 7 8 3 3 3 5 5 7 8 5 6 4 4 4 5 5 7 8 5 6 5 5 5 7 7 5 6 7 8 6 6 6 8 8 6 5 8 7 7 7 7 5 5 7 8 5 6 8 8 8 6 6 8 7 6 5 Table £Fig. 3. Examples of semigroups E and F that are not ideal nor Schreier extensions of A by S.
2.2 Semigroup extensions in general
From the examples given in Section 2.1 it is clear that, besides ideal and Schreier
extensions, there are several ways in which a semigroup can be constructed from two given semigroups. As a general definition of a semigroup extension we propose Definition 1
A semigroup extension (£, 0) of a semigroup A by a semigroup 5 is a semigroup E containing a subsemigroup A isomorphic with A, together with a congruence 6 on E such that ^ is a ö-class and that the factor semigroup £/ö is isomorphic with 5.
Since we shall discuss only semigroup extensions of a semigroup ^4 by a semigroup 5 we shall henceforth save repetition by omitting the qualification "semigroup" and write "extension (£,ö) of A by 5 " .
Definition 1. Also group extensions are covered by Definition 1 because group exten-sions are Schreier extenexten-sions (see Rédei's Theorem in Section 1.2).
From Definition 1 follow several properties of semigroup extensions in general. We shall develop these and introduce appropriate notions as we go along.
Throughout this chapter we adhere to the following notational convention. Let (£, 6) be an extension of yl by 5 and let >4 be a subsemigroup of £ such that .4 ^ ^4 and ^ is a ö-class (Definition 1 ensures that such an A exists). By a we denote the isomorphism of ^4 onto A so that A = Aa. The isomorphism of the factor semigroup £/ö onto 5 is denoted by P and the homomorphism Ö„a,j9 of £ onto 5 is denoted by y. Since ^ is a ö-class we have A6„^fP = Ay is an element of 5, we denote this ele-ment by /''. Notice that 6 = yoy"^ so that /^v~' = ^.
THEOREM 1. Let A and 5 be semigroups. There exist extensions of ^4 by 5 if and only if 5 contains an idempotent element. More precisely,
(i) Let (£, Ö) be an extension of y4 by 5 and let A be a subsemigroup of £ such that A = A and ,4 is a ö-class. Then Ay = i'* is an idempotent element of 5.
(ii) Let 5 contain an idempotent element /. Consider the direct product A ® S of A and 5. Let the congruence 0^ on ^4 ® 5 be öj = {{{a,s),{b,s)): a,b e A and J e 5 } . Then (A ® 5, 6g) is an extension of A by 5.
Proof. The first part of the theorem is an immediate consequence of statements (i) and (ii), so we proceed to show that (i) and (ii) are valid.
(i) Since A is a subsemigroup of £, clearly A A E A. Moreover ^4 is a ö-class, hence A is an idempotent element of the factor semigroup £/ö. Since £/ö = E6„^f and P is an isomorphism of £ö„3, onto 5, we have A6„^tP = A y = i'^ is an idempotent element of 5.
(ii) The mapping a of ^ into ^ ® 5, defined as aa = (a,i) for every a in A, is an isomorphism of ^4 onto the subsemigroup A = {(a,/): aeA} of A ® S. It is evident that 6g is a congruence and that A is a ö^-class. The mapping p of (A ® S)/ds onto 5, defined as {{a,s): aeA}P = s for every ö^-class of A ® S, is clearly an isomorphism. Hence (A ® 5, 6g) is an extension of A by 5.
Herewith ends the proof.
Notice that in Theorem 1 (ii) the congruence Og is the least congruence such that A is a congruence class.
In an extension (£, 6) of ^4 by 5 the subsemigroup ^ ^ ^4 of £ need not be unique and we have
THEOREM 2. Let (£, Ö) be an extension of v4 by 5 and let A be a subsemigroup of E such that A ^ A and ^ is a ö-class. If £ contains a subsemigroup A' ¥= A such that A' ^ A and A' is a ö-class, then A' n A = 0 and A'y = i^' ^ Z^.
Proof. Since A and A' are both ö-classes and A' # A, it is clear that A' n, A = 0. From the proof of (i) of Theorem 1 it is evident that both Ay = i'^ and A'y = i^ are idempotent elements of 5 and, since A' ^ A, also i"* # /^. Herewith ends the proof. Clearly, for an ideal extension of ^4 by 5 the idempotent element /^ is the zero ele-ment of 5, whereas for a Schreier extension i'^ is the identity eleele-ment of 5. Hence for these two types of extensions the element /^ of 5 is unique and also the subsemigroup ^ of £ is unique. From this it follows that for these two types of extensions Theorem 2 is vacuous. However, for an extension (£,ö) of A by S the semigroup £ may contain more than one subsemigroup which is isomorphic with A and which is also a ö-class. An example of this situation is provided by Figure 3 in Section 2.1. There semigroup F contains the subgroups G = {1,2} and G' = {5,6}. These are both isomorphic with A and both are ö-classes. The homomorphism y = 9„^fP of £ onto 5 maps G onto eg and G' onto t. Notice that £also contains the subgroup G" = {5,7} which is isomorphic with A but which is not a ö-class.
In connection with these remarks we introduce the following notions.
Definition 2. Let (£,Ö) be an extension of A by 5. For every subsemigroup ^ of £ such that A ^ A and y4 is a Ö-cIass we call the idempotent element /^ = Ay the extension idempotent of S with respect to A and (£,ö) is called an i^-extension of AbyS.
Now we turn to another consequence of Definition 1. On the semigroup £ exists the congruence Ö for which ^ is a ö-class, that is, Ö is an A-congruence. From Section 1.3 follows now immediately.
THEOREM 3. Let (£,ö) be an extension of A by S and let J4 be a subsemigroup of £ such that A = A and Ö is an A-congruence. Then the set C(A) of all A-congruences on £ is a complete lattice with respect to the operations n and v on congruences on £. The infimum yu and the supremum v of C{A) are given by
p = ^T, that is, n is the transitive closure of <^, where
^ = {(a,b) e ExE: a = b or else a, b e xAy for some x,ye £ ' } , and
V = {(a,b)eExE: xayeA if and only if xby e A for all x , > ' e £ ' } . Moreover, H Z 6 ^ V, where the congruence inclusion E is the partial ordering of the lattice C(A).
For ease of reference we now introduce the notion of extension lattice.
Definition 3. Let (£,ö) be an extension of A by 5 and let A be a subsemigroup of £ such that A = A and Ö is an A-congruence. The extension lattice of E with respect to A is the complete lattice C(A) of all A-congruences on £.
extension (£, ö) of A by 5 may contain more than one subsemigroup A ^ A such that
0 is an A-congruence, hence (£, Ö) may have more than one extension lattice. However,
for an ideal or a Schreier extension there is precisely one extension lattice. The dis-cussion in Section 2.1 makes it clear that both for an ideal and for a Schreier exten-sion (£, Ö) the A-congruence Ö on £ is the infimum of C(A). The same holds for 6s of the extension (A ® 5, Ö5) of A by 5 described in Theorem 1.
Consider a semigroup 5 containing an idempotent element /. Identifying / and the one-element subset {;} of 5 we see that / is a subsemigroup of 5. Moreover, the iden-tity relation i on 5 is a congruence such that / is a t-class, that is, t is an /-congruence. Furthermore, the factor semigroup S/i = 5. Hence (5,i) is an extension of / by 5. The complete lattice C{i) of all /-congruences on 5 has i as infimum. Theorem 3 provides an explicit characterization of the supremum of C(/). The preceding argu-ment is valid for any idempotent eleargu-ment of 5. Therefore we have the following property.
THEOREM 4. Let 5 be a semigroup and let / be an idempotent element of 5. The extension lattice C{i) of 5 with respect to / has the identity relation i on 5 as infimum and its supremum is the /-congruence {(s,t)e SxS: xsy = / if and only if xty = i for all X, ƒ 6 5 ' } .
There is an interesting relation between the extension lattices of the Theorems 3 and 4 as we shall describe in Theorem 5. However, it is expedient to give first a lemma which is useful not only for proving Theorem 5, but also in the discussions of Sec-tion 2.3.
LEMMA 1. Let A be a non-empty set of a semigroup 5, let cr he an element of the lattice C{A) of A-congruences on 5, and let /(a) be the principal A-ideal of C(A) generated by a. Furthermore, let fl be a non-empty subset of a semigroup T, let r be an element of the lattice C{B) of fi-congruences on T, and let /(T) be the principal
A -ideal of C{B) generated by r. Then there exists an isomorphism S of SeT„^^ onto
Tr„^i such that A<7„^,ö = fitn^, if and only if there exists a lattice-isomorphism cp of I{(T) onto /(T) such that for each a' in I((x) there is an isomorphism ö' of 5ff'n^, onto T((T'(p)n.i with A<7'„„^' = B(a'(p)„,,.
Proof. Assume (p is a lattics-isomorphism of /(o) onto /(T) such that for each a' in I(ff) there is an isomorphism S' of Sa'„^, onto T(a'(p)„^^ with A(T'„^J Ö' = B(a'(f))„^^.
Clearly, (p maps the infimum a of I((T) onto the infimum T of/(T), that is, atp = T,
so that T{(T(p)„^^ = Tr„^f. By assumption there is an isomorphism Ö of 5(T„3, onto
T((T(p)„^i = T'tna, with A(T„.j,S = fiT„a,. Herewith one half of Lemma 1 is proved.
Conversely, assume S is an isomorphism of S(T„^^ onto TT„^^ such that A(T„3,Ó = = fiT„3,. We have to show that there is a lattice-isomorphism (p of 1(a) onto / ( T ) such
^<^nai"' = ^('^'<P)nai- The proof, giving (p and ö' explicitly in terms of a, z, ö and a', is straightforward and proceeds by the following steps:
a) Define for an arbitrary element a' in I((T) a relation <j'(p on Tand show that (j'<p is an element of / ( T ) .
b) Show that ip is a one-to-one order-homomorphism of 1(a) into / ( T ) . c) Show that (p maps 1(a) onto / ( T ) .
d) Show that (p~' is a one-to-one order-homomorphism of I(r) onto 1(a). e) Conclude that cp is a lattice-isomorphism.
f) Define for an arbitrary element a' of 1(a) a mapping ö' of Sa'„^f into T(a'(p)„^, and show that Ö' is an isomorphism and Aa'„^,ö' = B(a'(p)„^i.
In following the details of these steps it may be helpful to refer to the diagram: Ö
A(j„„ e 5ff„3, y Tt„^, 3 BT„,,
''^nat ^ n a t A c s T^ B Now we give the steps a) t h r o u g h / ) .
a) Let a' he an element of 1(a), that is, a' is an A-congruence on 5 and a' 3 a. Con-sider the relation a'cp on T defined as
a'(p = {(t„t2)eTxT: tiX„^^ö'^a;^la'„^, = ?2T^nat'5"V„"3,'cr;,3,}
It is easily verified that a'tp is an equivalence relation on T, and even a congruence because T„a„ (5"', (Tn;,! and (Tnj,, are homomorphisms. From the assumption it follows that for the subset B of F holds BT„^^ó~^a^^l is just the subset A of 5. Now, a' is an A-congruence on 5, hence a'cp is a 5-congruence on T. Further-more, if (?i,/2) 6 "i^, then /jTna, = /2'^nat, thus (/],/2) e o^V, so that T ^ cr'ip. Hence a'(p is an element of I(r).
b) Let o'" be an element of 1(a) such that a" # CT'. From the definition of (p in step a) it is evident that a"(p ^ a'cp. Hence cp is a one-to-one mapping of 1(a) into I(x). Now, let a" e 1(a) be such that a" c a', If for two elements ti and tj of 7 holds (/i,/2) e ff'V then it follows immediately from the definition of (p in step a) that (ti,t2)e a'<p. Hence (p is a one-to-one order-homomorphism of 1(a) into / ( T ) . c) Let T' be an element of/(T), that is, T' is a fl-congruence on Tand T' 5 T. Consider
the relation x'tj/ on 5 defined as
T'"/' = { ( j l , J 2 ) e 5 ' X 5 : JifXna.^Tn'a'^a, = J2<^„a,^T;a? < a . }
-As for ff'(p in step a) one can verify that T'I/^ is an element of 1(a). Moreover, from step a) follows immediately that z'{j/(p is an element of I(x). Now, for ti,t2 e T holds (ti,t2) e x'lj/q) if and only if
that is, if and only if (t^,t2)e x'. Hence x'ij/(p = x'. Herewith we have shown that each element of I(x) is the image of an element of 1(a) under the mapping (p. Taking this together with the results of step b) we have established that <p is a one-to-one order-homomorphism of 1(a) onto I(x).
d) Moreover, since x'lj/cp = x' for every element x' of I(x), we have \j/ = <p~^ and ij/ is a one-to-one mapping of l(x) onto /(cr). Suppose x" is an element of I(x) such that T" ^ T'. If for two elements s^ and ^2 in •S' holds (^1,^2) e t^ ""A, then it follows immediately from the definition of i// in step c) that (^,,^2) e ^'A- Hence ij/ = <p~^ is one-to-one order-homomorphism of I(x) onto 1(a).
e) Taking the results of the steps c) and d) together we conclude that 9 is a lattice-isomorphism of 1(a) onto l(x).
f) Let a' be an element of 1(a) and let s be an element of 5 so that sa'„^t is an element
of 5crn^,. Consider the subset sa'^^ia'„~i^a„._,,Sx~,,l of T. From the argument in step a) it is evident that this subset is precisely one a'cp-dass of T, that is, an element of
T(a'(p)„,,. Let Ö' he the mapping <T;;,'(7„„^T„l,'(ff'^)„„ of 5o-;„ into T(a'(p)„^,. Since
ffJiai, CTnat' ^ ^"^ Tn^, arc homomorphisms, ö' is clearly a one-to-one mapping of
Sa'„^i onto T(a'(p)„.,i and also a homomorphism. Moreover, the element Aa'„^, of Sa'„„f is mapped by ó' onto the element B(a'(p)„^i of T(a'(p)„^i. Hence ö' is an
iso-morphism of Sa'„^i onto T(a'(p)„^i and AcrJ,^,^' = fi(o-'9)„j„. Herewith ends the proof.
For ease of reference we now introduce the notion of isolamorphism.
Definition 4. Let A be a non-empty subset of a semigroup 5, let a be an element of
the lattice C(A) of A-congruences on 5, and let 1(a) be the principal A -ideal of C(A) generated by a. Furthermore, let B be a non-empty subset of a semigroup T, let x
be an element of the lattice C(B) of iB-congruences on T, and let I(x) be the principal
A-ideal of C(B) generated by x. An isolamorphism cp: 1(a) -> I(x) is a lattice-iso-morphism of 1(a) onto l(x) such that for each a' in 1(a) there is an isolattice-iso-morphism
S' of Sa'„.^^ onto T(a'(p)„.^, with Aa'„,J' = B(a'(p)„.^,.
We shall need the following remark in Section 2.3. Lemma 1 states that a mapping
(p of 1(a) onto / ( T ) is an isolamorphism if and only if there exists an isomorphism ö of Sa„^t onto Tx„^i such that Aa„.^,ö = Bx„._,^. But then (5"' is an isomorphism of £T„3,
onto 5cr„j, such that Bx„^^è~' = Acr„^,. Hence <p"': / ( T ) ^ /(cr) is an isolamorphism. Moreover, let B' he a non-empty subset of a semigroup 7", let x' be an element of the lattice C(B') of ^'-congruences on T', and let I(x') be the principal A -ideal of C(B') generated by x'. \f(p': 1(a)-* l(x') is an isolamorphism then clearly <p~V- ^(T)->/(!') is an isolamorphism.
Returning now to the relation between Theorem 3 and Theorem 4, alluded to be-fore stating Lemma I, we have
THEOREM 5. Let (£, Ö) be an /''-extension of A by 5, let C(A) he the lattice of A-con-gruences on £ and let /(Ö) be the princiapl A -ideal of C(A) generated by Ö. Further-more, let C(/^) be the extension lattice of 5 with respect to i'^. Then there is an iso-lamorphism (p of C(i'^) onto 1(6). Moreover, cp is given, for all g in CO"*), by
g(p = {(a,b e £ x £ : (ay,by) e g},
and the isomorphism ö': Sg„^i -» E(g(p)„^i is given, for all sg„^^ in 5gna„ by •y^nac^' = {aeE: aye
i'gnatgnaJ}-Proof. We verify that the conditions of Lemma 1 are satisfied. Clearly, /^ is a subset of 5, the identity relation i is the infimum of C(i'^) so that C(/^) is the principal A -ideal of C(i^) generated by i. Furthermore, A is a subset of £, Ö is an element of C(A) and /(Ö) is the principal A -ideal of C(A) generated by Ö. The isomorphism j8~' of 5 = 5/i onto £ö„3, is such that /^J?~' = Aön^,. Hence it follows immediately from Lemma 1 that there is an isolamorphism cp of C(/^) onto /(Ö). From step a) in the proof of Lemma 1 it follows now that for an element g of C(/^) the element g(p of C(A) is given by
g(p = {(a,b) e ExE: a9„^^Pg„^^ = b9„Jg^^,} or, since 9„^,P = y,
g(p = {(a,b) e ExE: (ay,by) e g}.
From step f) in the proof of Lemma 1 it follows that for an element sg^.^^ of Sg„^^ the element sgn,,^6' of E(g(p)„.^i is given by
•ïönat^' = •ïenaten~aNj8"'ö~'(g<P)„at = ^Qn^lQ^^lj'\Q(P)n^i = = {a 6 £ : ay e Jg„a,g„l,t }•
Herewith ends the proof.
2.3 The set of extensions of A by S
Let A and 5 be two semigroups and let 5 contain at least one idempotent element. By (f we denote the set of all extensions of A by 5. For each idempotent element / of 5 we denote by Si the set of all /-extensions of A by 5, that is, é°i is the set of all exten-sions of A by 5 with / as extension idempotent. Theorem 1 ensures that the extension (A ® 5, 9s) is an element of <f, for each idempotent element / of 5. Hence the set ^, is not empty and every pair of these sets has at least one element in common. Theorem 5 entails, among else, that the extension lattice of each element of <f; has a principal A -ideal which is isolamorphic with the extension lattice C(i) of 5 with respect to /. On the basis of Theorem 5 it is possible to endow the set ^i with some structure and we proceed to develop this.
Definition 5. Let A and 5 be two semigroups, let 5 contain an idempotent element /, and let Si be the set of all /-extensions of A by 5. Let (£,ö) and (£',0') be two ele-ments of <f j, and let C(A) and C(A') be their extension lattices, respectively. On #, we define the equivalence relation a by ((£,0), (E',9'))e(ü if and only if there is an isolamorphism (p of C(A) onto C(A'). The set of all /-extensions of A by 5 which are cü-equivalent with (£,ö) is denoted by [£,ö].
It is easily verified that CD is indeed an equivalence relation on S^. Notice that, according to Lemma 1, we may define co also as ((£,0), (£',0')) e to if and only if there an isomorphism ö of Ep„^i onto f'/i^i such that A^„^iö = A'/^óai, where fi and ^' are the infima of C(A) and C(A)'/, respectively.
If there are ideal and/or Schreier extensions of A by 5, then these are co-equivalent with the extension (A ® 5, 9g). This is so because for each such extension (£,ö) there is an isomorphism p mapping E9„^i onto 5 with A9„^iP = i^ and because 9 = p. Furthermore, if (£,ö) is an extension of A by 5, then all extensions (E',9') of A by 5, such that £ ' is isomorphic with £, are oj-equivalent with (£,ö). These remarks make clear that, although we have based the equivalence relation m on S^ on the somewhat abstract content of Lemma 1 and Theorem 5, it is not unnatural to consider the relation co.
On the factor set êJ(o we now define a partial ordering.
Definition 6. Let A and 5 be semigroups, where 5 contains an idempotent element /, and let <f; be the set of all /-extensions of A by 5. Let [£,ö] and [£',0'] be two elements of the factor set ^./o). Define on Si/w the partial ordering -< as [£,ö] -< [£',0'] if and only if there is a homomorphism ;; of £^„3, onto £'/inat such that Ap„^^r] = A'p'„.^^.
That -< is indeed a partial ordering is readily verified. Notice that t] is an isomor-phism if and only if [£,Ö] = [£',0'], according to the remarks following Definition 5. Using Lemma 1 once more, we have
THEOREM 6. Let A and 5 be two semigroups, where 5 contains an idempotent ele-ment /, and let <?,• be the set of all /-extensions of A by 5. Let [£,ö] and [£',0'] be two elements ofSJco. Then [£,ö] -< [£',0'] if and only if C(A) contains an element n such that there is an isolamorphism of the principal A -ideal l(n) of C(A) generated by 71, onto C(A').
Proof. Assume [£,ö] -< [£',0'], that is, there is a homomorphism rj of Ep„^, onto £'/iJ,a, such that Ap„^,ri = A'/i^^t where p and /i' are the infima of C(A) and C(A'), respectively. Let n he the A-congruence on £ induced by the homomorphism ^„a,>7, that is, 7c = ^„a,»7o(^„a,J7)"'. Then the mapping Ö defined, for each element an„^f of £7r„a„ with aeE, as a^t„^^S = an„^in~J^fi„^,ri, maps En„^^ onto £'/<naf Moreover, it is easy to verify that S is an isomorphism and that A^t„^^S = A'/zóa,. Hence Lemma 1 ensures that there is an isolamorphism of I(n) onto C(A').
Conversely, assume there is an isolamorphism (p of l(n) onto C(A'). According to Lemma 1 there is an isomorphism S of £7tnat onto £'|U^a, such that A7t„a,(5 = A'p'^^^. Now 71 is an element of C(A) so that n ^ fi. Hence the homomorphism ij/ = //„aNnat of £yUna, outo £7r„a, is such that Aju„a,i/' = An„^f. Furthermore, [f/S is a homomorphism of £/^„3, onto E'p'„^i such that Ap„^iil/ó = A'^u^^t, so that indeed [£//i] -< [£'/;i']. Herewith ends the proof.
Another property concerning the partial ordering -< on the factor set ^,/co is given by
THEOREM 7. Let A and 5 be two semigroups, where 5 contains an idempotent element /, and let Si he the set of all /-extensions of A by 5. Let [£,ö) and [£',0'] be two elements of SJo). If there is a homomorphism C of £ onto £ ' such that AC = A', then [£,ö] -< [£',0'].
Proof. Let p' be the infimum of the extension lattice C(A') of (E',9') with respect to
A'. Consider the relation 7: on £ defined as 7r = {(a,b)e ExE: flC/^I,a, = bCp'„^i}. One easily verifies that 7t is an A-congruence on £, so that n is an element of C(A). Furthermore, the mapping 5 = TTM'C/^nat 'S an isomorphism of En„^i onto £'^i,at and
An„.^^ö = A'p'„^f. Invoking Lemma 1 we conclude that there is an isolamorphism of
the principal A-ideal l(n) of C(A) generated by n onto C(A'). From Theorem 6 follows now immediately [£,ö] -< [£',0']. Herewith ends the proof.
Notice that, in case ^ is an isomorphism, we have [£,Ö] = [£',0'] according to the remarks following Definition 5. Considering the partially ordered set SJco of sets of co-equivalent /-extensions of A by 5 one can easily indicate its supremum.
THEOREM 8. Let A and 5 be semigroups, where 5 contains the idempotent element /, and let Sj he the set of all /-extensions of A by 5. The factor set SJco, partially ordered with respect to -<, contains [A ® 5, 9g] as supremum.
Proof. Let (E',9') he an arbitrary /-extension of A by 5. Then Theorem 5 states that
there is an isolamorphism cp of C(i) onto the principal A -ideal /(Ö') of C(A') gener-ated by Ö'. From Theorem 1 we know that (A ® 5, dg) is an /-extension of A by 5. Moreover, öj is the infimum of the lattice C(A) of all A-congruences on the semigroup
A (g) S. Hence, according to Theorem 5, there is an isolamorphism cp' of C(i) onto C(A). The remark following Definition 4 entails that the mapping (p^\' is an
isola-morphism of /(Ö') onto C(A). Hence [£',0'] •< [A 0 S, 0^] due to Theorem 6. Herewith ends the proof.
Examples of most of the properties derived in this section are furnished by the extensions £ and £ of A by 5 given in Figure 3 of Section 2.1. In Figure 4 we give the extension lattice C(eg) of 5 with respect to Cg, the extension lattice C(A) of £ with respect to A = {b,c}, and the extension lattice C(G) of £ with respect to G = {1,2}. In each case the congruences are given in Figure 4 by their congruence classes.
.: {.,}, {5}, {/}, {«} g,: {eg}, {s}, {t,u} Qi- {es}, {s,u}, {/} Qi'- {es}, {s,u,t}
a: {b,c}, {d,f}, {g}, {h} a,: {b,c}, {d,f}, {g,h} C72: {b,c}, {d,f,h}, {g} <^3- {b,c}, {d,fg,h}
Extension lattice C(es) Extension lattice C(A)
/^ 6 öi Ö2 Ö3 ö, Ö5 06
e,
{1,2}, {3,4}, {5}, {6}, {7} {1,2}, {3,4}, {5,6}, {7,8} {1,2}, {3,4}, {5,6,7,8} {1,2}, {3,4,7,8}, {5,6} {1,2}, {3,4,5,6,7,8} {1,2}, {3,4}, {5,8}, {6,7} {1,2}, {3,4,6,7}, {5,8} {1,2}, {3,4}, {5,7}, {6,8} {1,2}, {3,4,5,7}, {6,8} {8} Fig. 4. Extension lattice C(G) Example of extension lattices.It is easily verified that S ^ S/i ^ E/a ^ F/9 ^ F/Ö4 ^ F/ög, S/g, ^ E/a, ^ £/Ö,, 5/g2 ^ £/c72 ^ F/Ö2 ^ £/Ö5 s F/Ö7, and 5/g3 ^ E/a3 ^ f/öj. Furthermore, Cg, the congruence class A of £ and the congruence class G of £ are mapped onto each other by these isomorphisms. From this it follows that C(eg) is isolamorphic with C(A) and also with the principal A -ideals /(Ö) and /(ÖJ of C(G) generated by Ö and Ö4, respect-ively, just as Theorem 5 ensures. But C(es) is not isolamorphic with the principal
A -ideal 1(9^,) of £ generated by 0^ because F/9(, ^ 5. However, there is clearly a lat-tice-isomorphism of C(es) onto /(ög). The ej-extension (E,a) of A by 5 is co-equivalent with the ej-extension (A ® 5, 9g) of A by 5, hence [£, a] = [A 0 S, öj] so that, according to Theorem 8, [£, cr] is the supremum of S^Jw with respect to the partial ordering -<. Consider the mapping rj of F/fi onto £/Ö- given by {1,2}/? = {b,c}, {3,4}r, = {d,f}, {5}ri = {6}ri = {g}, {7}j? =_{8}f? = {h}. Clearly, ri is a homomor-phism and since {1,2}>; = {b,c}, i.e. Grj = A, we have [£,ö] -< [E,a], according to Definition 6. As Theorem 6 ensures, the principal A -ideal /(Ö) of £ generated by Ö is indeed isolamorphic with the extension lattice C(A) of £ with respect to A.
Now consider the mapping C of £ onto £ given by IC = b, 2C = c, 3^ = d, 4C = ƒ, 5^ = 6C = g, 7C = 8C = h. Clearly C is a homomorphism and {1,2}^ = {b,c}, i.e. GC = A, so that Theorem 7 ensures also that [£,ö] -< [E,a].
