Proper P-compatible hypersubstitutions

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B r u c e B e m d t University of Illinois Illinois, USA K l a u s Denecke Potsdam University Potsdam, Germany W i l l e m Fouche Univ. of Pretoria R. of South Africa S . R . L o p e z - P e r m o u t h Y o n g w i m o n L e n b u r y Ohio University Mahidol University Ohio, USA Bangkok, Thailand Nguyen T o N h u K . Oshiro New Mexico State University Yamaguchi University New Mexico, USA

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G a r y F . B i r k e n m e i e r Uni. Southwestern Louisiana Lafayette, Louisiana, USA N g u y e n V i e t D u n g Institute of Mathematics Hanoi, Vietnam

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The Ohio State University Lima, Ohio, USA

J . B . Srivastava Indian Inst, of Technology Delhi, India

T r a n D u e V a n Institute of Mathematics Hanoi, Vietnam

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E A S T - W E S T J O U R N A L O F M A T H E M A T I C S Department of Mathematics, Facility of Science Khon Kasn University, Khon Kaen 40002, T H A I L A N D

P R O P E R P - C O M P A T I B L E

H Y P E R S U B S T I T U T I O N S

K. Denecke* and K . Mruczek*

University of Potsdam, Institute of Mathematics Am Neuen Palais, 14415 Potsdam, Germany

e-maiUkdenecke@rz.uni-potsdam.de ^ Institute of Mathematics, University of Opole

ui Oleska 48, Opole, Poland

e-moil: mruczek@math.uni.opole.pl

Abstract

An identity s *s t is called a hyperidentity in a variety V if by sub-stituting terms of appropriate arity for the operation symbols in s « t, one obtains an identity satisfied in V. If every identity in V is a hyper-identity, the variety V is called solid. All solid varieties of a given type T form a complete sublattice <S(T) of the lattice £(r) of all varieties of type r . The concept of an M-solid variety generalizes that of a solid variety. An equation s as t of terms of type r is called P-compatible where P is a partition of the set F = {fi\i £ / } of operation symbols of type T if it has the form n « Xi or /;(ti, -. . , t „() fj{t\, • • • ,t'n.)

with fj € [fi]p , where [fj\p is the block of P containing fj. A variety is called compatible if it contains only compatible identities. All P-compatible varieties of type T form also a sublattice of the lattice of all varieties of type r. We ask for the intersection of both lattices, i.e. we want to characterize solid varieties which are P-compatible or M-solid varieties which are P-compatible.

1 Preliminaries

Our informal definition of a hyperidentity shows that we are interested in a map which associates to each ni-ary operation symbol fa an n;-ary term

o{fi)-K e y words and phrases: P-compatible identity, hyperidentity, hypersubstitution. (1991) Mathematics Subject Classification: 20M14, 20MO7.

34 Proper P-compatible Hypersubstitutions

Any such map is called a hypersubstitution. Let WT(X) be the set of all terms

of type r on an alphabet X = {xi,x2,... ,xn,...}. Using a hypersubstitution

a we can define a uniquely determined mapping a defined on terms by

(ii) &[fi{tu.\.,tni)] := ff(/4)(d[ti],...,ff[tnil).

By Hyp(r) we denote the set of all these hypersubstitutions. If we define a multiplication oh on the set Hyp(r) by <T] oh 02 := ffi O<T2 where o is the usual

composition of functions, together with ff«i(/t) : = fi(xi,.. .,x„t) we obtain a

monoid Hyp(r) = (Hyp(r); oh; <jid). If M is a submonoid of the monoid of all

hypersubstitutions of type r then an equation s ~ t of terms of type T is called M-hyperidentity in the variety V of groupoids if for all a £ M the equations a[s] « &[t] are satisfied as identities of V . Hyperidentities are M-hyperidentities for M = Hyp(r). A variety V of type r is called M-solid if each of its identities is an M-hyperidentity for M = Hyp(r). All M-solid varieties of type T form a complete sublattice SM(T) of the lattice £ ( r ) of all varieties of type T with

M i C M2= > 5M, ( T ) 2 SM l( r ) .

To test whether an identity s w t of a variety V is an M-hyperidentity of V our definition requires that we check, for each hypersubstitution in M , that a[s] ~ a[i\ is an identity of V. Indeed, we can restrict our testing to certain "special hypersubstitutions". We recall of two concepts, both introduced by J . Plonka ([4]).

Definition 1.1 Let V be a variety of type T . A hypersubstitution a is called V-proper if for every identity s ~ t in V, the identity a[s) « o[i] also holds in V. We use P{V) for the set of all V-proper hypersubstitutions.

It is clear that ( P ( V ) ; oh; oi d) is a submonoid of Hyp(r) = {Hyp(T); oh; aid)

and that a variety V is M-solid for M — P(V) and P(V) is the largest M for which V is M-solid.

Definition 1.2 Let V be a variety of type r. Two hypersubstitutions ai,a2

are called V-equivalent (o-i~v&i) if &i(fi) « <*i(fi) <""e identities in V for all i€l.

This relation can be extended to arbitrary terms t, i.e. tri~v^2 &i[t} « (72[t]

is an identity in V. Then one can prove: If ffi~v02 and di[s\ « &i[t] then 02\s] ~ oafi] is an identity in V.

We will use the following denotations:

Id V - the set of all identities satisfied in the variety V, Cp{r) - the set of all P-compatible equations of type r ,

Cp{V) = CP(T) n Id V - the set of all P-compatible identities of V,

EX(T) - the set of all externally compatible equations of type T , i.e. in-compatible for P = {{fi}\i G I } ,

Ex{V) = EX[T) n Id V,

N(T) - the set of all normal identities of type r , i.e. P-compatible for P = { { / * } ! * € / } ,

N(V) = N(T) n Id V.

It is easy to see that C p ( r ) and Cp(V) are equational theories, i.e. closed under the rules of consequences for identities.

2 Cp(V)-proper hypersubstitutions

Definition 2.1 J4 hypersubstitution a G Hyp(r) is called Cp(V)-proper if for alls Pit G CP(V) we have &[s] w € C p ( V ) ( i.e. <r[,s] ss ff[t] € /rf V and

a[s] w <r[f] = Xi <E. X or ex(a[s\) G [ex(<r[i])]p where ex(a[i\) denotes the first operation symbol occurring in the term a[t\ ) .

Let McP{V) be the set of all Cp(V)~ proper hypersubstitutions of type T .

Then we have

L e m m a 2.2 McP{V) forms a submonoid of Hyp(r).

Proof. If a sa t € CP{V) then did\s] ai ( i[i] G Cp(lO> thus

cr.d G M c p ( V ) - If <7i,cr2 G MCp{V) then for all 5 a £ G C F ( V ) we

have (72[s] « <72fl G C P ( V ) and then <7i[<r2[s]] ~ oi[°i\t\\ € CP{V) , i.e.

(ffi °h ffajT [*] « (ffi °ii <72)~ [t] G CP(V). Therefore 0\ oh cr2 G MCp(V). •

Remark that there are different possibilities to define sets of hypersubstitu-tions which are connected with P-compatible identities of the variety V. For instance we could also define a hypersubstitution to be Cp(V)-generating if for all s « t G Id V it follows that a[s] « a[t] G CP(V). If we denote by GCp(V)

the set of all Cp (V)-generating hypersubstitutions we have

L e m m a 2.3 GcP{V) is a semigroup of hypersubstitutions which in general is

36 _{Proper P-compatible Hypersubstitutions}

Proof. If 0-1,(72 G G o ( V 0 then for all s sa f e Id V we get &2[a] ss

*aM € C P ( V ) and thus «Ti[o2[s]] W *i[*9[f]] G C j » ( V ) . This means G cP( V " ) i s

closed under the product oh. But in general GcP{V) is not a monoid since

s « t G Id V , but s ss t £ C p ( V ) and then <rid[s] SS o-id[t] £ Cf( V ) . •

R e m a r k s :

1. If V is an idempotent variety, (i.e. {fi(x,...,x) = x) G Id V ) then a hypersubstitution belonging to GcP(V) has to map each ft to one of the

variables x\t...,x„t.

2. Clearly, GcF{V) is a subsemigroup of the monoid P{V) of all proper

hypersubstitutions of type r and McP(V) is the monoid of all proper

hypersubstitutions of the variety VcP •= Mod(Cp(V)) which is denned

by all P-compatible identities of the variety V.

That means, the variety VcF is M-solid for the monoid McP(V) and

McP(V) is the greatest monoid of hypersubstitutions such that VcP is

M-solid .

T h e o r e m 2.4 Let V be a variety of type r and let P be a partition of the set {fi\i G / } of operation symbols. Let McP(V) be the monoid of all CP

(V)-proper hypersubstitutions. If McP{V) = Hyp{r) then P — {{fi}\i G / } or

P = {Miei}.

Proof. Let s « t be an arbitrary identity of Cp{V) and assume that P ^ G / } . Then we can assume that s = / j ( s i , . . . , s„<) and t = fj{t\,... ,tnj) with fj G [fi]p,fi ^ fj- (Such an identity exists since

P G / } ) . Consider now a hypersubstitution which maps fi to fi(xu... ,xni) and fj to ft(/,( ) , . . . ,fj{xi,. •. , xn j. ) ) , where h is an

arbitrary operation symbol of P = {fi\i G / } . We may assume that h is not miliary, otherwise we change the role of / and g. Since Hyp(r) = Cp{V) we

obtain &[s] « a[t] G CP(V) and h G [fi]P and P - {fi\i G / } . •

Note that Theorem 2.4 is a reformulation of [2,Theorem 8] which says that if Mod{CP(V)) is solid and Mod{CP{V)) ^ Mod{Ex{V)) then Mod(CP{V))

is normal. The proof is also only a reformulation of the proof of (2,Theorem 8j. We consider some more examples. We will call a hypersubstitution <7 of type T a pre-hypersubstitution if for every i 6 I the term is not a variable ([3]). Let Pre(r) be the set of all pre-hypersubstitutions of type r . Let T be the trivial variety of type r , i.e. T = Mod{x & y) and let Id X be the set of all

identities of type r . Therefore CP{T) = CP{T) and MCP{T) = Pre{T)C\{a\U e

[fj\p ex(<j{fi)) e \ex{a(fj))}p}. If V = Alg(r) is the class of all algebras

of type r , i.e. V = Mod{x ^ x} then V is solid. The set Id V consists of all equations where the terms on the left and on the right hand side are the same. Clearly, Cp{V) = Hyp(r) for any partition P of {fi\i 6 / } .

Since McP (V) is a monoid we can apply the theory of M-hypersubstitutions

and M-solid varieties developed in [1). We can apply hypersubstitutions o~ € Mcpiy) to both, to equations and to algebras. If s « t is an equa-tion of terms of type r then we can form o[s] a[t]. and define an operator #Cp by

- {&[*] » € MCp(V),s « f € E , E C WT( X )2} .

The application of hypersubstitutions to an algebra ¿4 = (J4; {fiA)i£i) of type

r is defined by

* C p l # ] := {*\M\<> € MC, ( V ) , J 4 e C Alg(r)}, where :=

( 4 ( f f ( / 0A) < € / ) .

It is easy to see that both operators have the properties of closure operators which are defined for arbitrary non-empty sets as union of the results which we obtain if we apply them to one-element sets, i.e.

*2,[E1 = U

^

[if] = u *c„(U})

Such operators are called addit.ive. Further they are connected by the prop-erty

s M * e Id 4» a « *}) € Id If.

Because of this property we speak of a conjugate pair of additive closure oper-ators.

Further we use the following denotations:

HMC (i/) Id ( V ) - the set of all Mcv(V)-hyperidentities satisfied in the variety

V and

HMCp(v)Mod(Y!,) - the class of all algebras of type r , such that every equation

of is an Mc7p(V)-hyperidentity of this algebra.

Further, we say that a variety K is McP{V) -solid if ^[K] = K. From

the properties of the pair {X^c (v)>^Mc (V)) M a conjugate pair of additive

closure operators we obtain the following characterization of MQP{V)-solid

38 _{Proper P-compatible Hypersubstitutions}

T h e o r e m 2.5 For all varieties K of type r and for all equational theories E of type T the following conditions (i)-(iv) and the conditions (i')-(iv') are equivalent:

(i) K — HMcp{V)ModHMCp(y)Id (K) (K is an McP(V)-hyperequational

class),

XMCp(v)W = K is MCp(V)-solid,

(in) X^v)[Id(K)] = Id{K),

(iv) Id (K)= HMcp{V)Id (K), and

(*') £ = HMcp{V)Id HMcp(V)Mod$:),

(iii') X&Cpiv)[ModC£)) = Mod(Z),

(m Mod(Z) = HMcp{v)

Mod(Z)-We have already mentioned that the variety VcP = Mod(Cp(V)) is McP

(V)-solid. Therefore VcP satisfies the equivalent conditions (i),(ii),(iii),(iv).

From the general theory (see [ 1)) it follows also that the class of all McP{vy

solid varieties forms a complete lattice which is a complete sublattice of the lattice of all varieties of type r .

3 P-compatible relations on hypersubstitutions

In analogy to the relation ~ v we define the following binary relation on the set Hyp{r) and on submonoids of ffyp(r).

Definition 3.1 Let V be a variety of type r and let P be a partition of the set of operation symbols {fi\f e / } ofV. Let Cp(V) be the set of all P-compatible identities satisfied in V. Then for any two hypersubstitutions <ri,o~2 6 Hyp(r) we define

We notice that ~cP(v) is an equivalence relation on Hyp(r) . It can be

easily-shown that for all terms t G WT(X), if o\ ~ cP( V ) °~i then di[t] w &2[t] G

CP(V).

The relation ~<jP(v) can also be restricted to the monoid McP

(v)-T h e o r e m 3.2 (v)-The monoid Mcp(v) 1 5 a union of full equivalence classes with

respect to the relation ~ o ( v )

-Proof. We have to show that if o\ G McP(v) and if a\ ~ cP( V ) " 2 then

o~2 G McP(v)- Indeed, u\ G MCp(v) means that for each s ~ t G Cp{V) the

identity <7i[s] w di[t] belongs also to Cp(V). The relation «72 ~ cP( V ) &i

im-plies <72[f] « o-j[t] 6 C P ( V ) for all t € WT{X). But then, by transitivity we get

cri[s] ~ #a[*] € CP(V), and this means CT2 G M Cp( V ) - D

Theorem 3.2 shows also that, if we want to check whether an identity is an Mc7p(v)-byperidentity, we can restrict our checking to one representative from each equivalence class with respect to ~ cp( v ) - We can also show

C o r o l l a r y 3.3 The restriction of the relation ~Cp(v) to the submonoid McP{v) is a congruence relation on the monoid McP

{v)-Proof. We show that the restricted relation ~cP(v) \McP(V) is a right and

a left congruence on McP{v)- Assume that o~i ~ C V ( V ) I MC <V) FFA *N A T

a € MCp(v)- Then for the term <r(/j) we have Oi\d{fi)) w c ^ C / i ) ] € C p ( V )

and therefore ofc <r ~ CP( V ) | M0 ( V ) "2 ° A ". From ax ~cPlv))„Cp(V) a2 1 1

follows <Ti(/j) w ff2(/i) G Cp{V) and for every <r G M cP( v ) also *[o-i(/i)] «

e C / > ( V ) , i.e. CTOfcff! ~C p ( Vr ) | MC p (v , f f ° hf fi € C P ( V ) . •

If we consider the class of the identity hypersubstitution, we notice that it forms a submonoid of McP{v) since if o~i ~ cP( v ) "id and c"i ~ C F ( V ) o~\d then

we have Oi(fi) as <rw(/i) = fi(xlt... ,xni) G CP{ V ) and <r2(/i) « <?id(/i) =

/ i ( r c i , . . . ,xnt) G C p ( V ) and then also (ax

ocr

){/j)

= ffiM/i)] w

C p ( V ) , i.e. (<TI oh <r2)(/i) = CTi{a2(/i)j * " i ( / , ) = o-a € C p ^ ) , that means

o\ oh a-i ~cP{V)

°~id-Finally we want to remark that the previous definitions and theorems can be generalized in the following way:

Let V be a variety of type r and let T(T) be the set of all identities of type r which fulfils a given property. For example the property could also be that

Proper P-compatible Hypcrsubstitutioas

the set of variables occurring on both sides of the identity agree, so that T ( r ) is the set of all regular identities of type r. We will also assume that T ( T ) is an equational theory. We set T(V) := T(T) n Id V and if E is an equational theory we set T ( E ) := T{r) n E .

Then we define a hypersubstitution to be T(V)-proper if for all s ft* * £ T(V) we obtain a[s\ « a[t] e T ( V ) .

Let Mj^v) t>e t r ie set of all T(V)-proper hypersubstitutions of type r . Clearly

MT(V) 1S a submonoid of Hyp(r) and we get a theorem similar to Theorem 2.6.

The relation ~ cP( V ) can be also generalized and we define

a ~nv)

0-2 Vi e /(<ri(/0 »«r

(/0 e r(V)).

Using this definition we obtain theorems similar to Theorem 3.2 and to Corol-lary 3.3.

References

[1] K . Denecke and M. Reichel, Monoids of hypersubstitutions and M-solid varieties, 117-126 in: Contributions to General Algebra 9, Wien, Stuttgart 1995.

[2] K . Denecke and K . Halkowska, On P-compatible hybrid identities and hy-peridentities, Studia Lógica 53 (1994), 493 -501.

[3] K . Denecke, Presolid varieties, Demonstrate Matemática, Vol.XXVII, 3-4(1994), 741 -750.

[4] J . Plonka, Proper and inner hypersubstitutions of varieties, Proceedings of the International Conference: Summer School on General Algebra and Ordered Sets, 1994, Palacky University Olomouc 1994,

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