Translations on ternary algebras

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ROCZN1KI POLSK IEG O TOWARZYSTWA M ATEM ATYCZNEGO Séria I: PRACE MATEMATYCZNE XXII (1980)

Juhani Nieminen (Oulu)

Translations on ternary algebras

1. Introduction and general observations. In [12] Poes considered translations on universal algebras generalizing Szâsz’s results given in [13], [14] and in [15]. The purpose of this paper is firstly to present a few observations on translations on universal algebras and thereafter consider translations on ternary algebras. Ternary algebras, in which we are interested, are considered by Nebesky in [4] and [5] and some further results are also given in [6], [7], [8], [9] and in [10].

The concept of an ideal in tree algebras defined by Nebesky [4] suggests us to use a different definition than Poes’; the concept of a translation is also differently defined because e.g. in simple ternary algebras Poes’ translations are identical mappings.

By .д/ = { A , . 9") we denote an algebra with an n-ary operation, n ^ 2,

<2 £ : ¥ , where . ¥ is the set of the operations on and A is the carrier

set of If {al , . . . , a n} Я A , then Q(at , ..., a„)e A, too. We assume that

for each x e A , x = Q(ax, ..., an) for some subset Ç A. A subset U of A is said to be an ideal of the algebra .я/, if x e U when

* = Q(a1, ..., a„) and at least n — 1 elements of the «-tuple (a1, . . . , a n)

belong to U.

A single-valued mapping 9 of the set A into A is called a Q-translation

of the algebra = <A, J 5" ) , Q e .'W, if for each «-tuple ( a j , . . . , ^ ) of A and for each i, 1 ^ i ^ «, the following formula holds:

9 ( Q ( a ^ ..., a,_!, af, ai + 1, ..., a„))

= Q ( 9 ( a i ) , ..., ah 9 ( a i + l) , ..., -9(an)). As we shall consider ^-translations only,/we shall call them briefly translations.

An endomorphism of the algebra sé is a mapping h from A into A such

that for each Q e 3F

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for any «-tuple (a1, . . . , a n) of A. An endomorphism h of sé onto its ideal U which leaves U elementwise fixed is a retraction and U is a retract

of ,sé. A function к on sé is called idempotent, if k(k(x)) = k{x) for each

x e A.

As the following example of Poes [12] shows, a translation of an algebra need not be idempotent. Let us consider the closed interval [ —1, 1] = A, where Q(at , ..., an) = a1a2 ... a„^1a„, {al 9 a„} ç= [ — 1 ,1 ], and let n = 2m for some integer m ^ 1. Clearly the mapping / : A -> A defined as follows: /( x ) = —x when x e A, is a translation on ( A, Q) , but / ( / (x)) = x ф —x = / (x). For idempotent translations one can be prove a generalization of [11], Corollary 1.

Theorem 1. An idempotent function f on sé — <A, J ^ ), Q e 9 , is

a translation on sé if and only if f is a retraction of s é .

P ro o f. Let / be a translation and an idempotent function on sé . If

y e f ( A ) , then y — f (x) for some x e A . Let aly ..., a„_t g/(A ) and let us consider an element Q(ax, ..., an_ 1, a„), where ane A. As a,- = / (yj) for some

y} e A and for each j, j = \ , . . . , n - l , Q(au ..., а„-и an) = Q( f ( y i),... . . . , f ( y n-i),a„) = f ( Q ( y i , . . . , y n- i , a „ ) ) e f ( A) . Thus f (A) is an ideal of sé

and it remains to show that / (A) is a retract of sé with respect to / . Let y e f ( A) . So V = Q(fiai ), ..,,/(< 1,- ,) , Now f(y)

=

Q ( f { f

M , . . . , / ( / (a,-,)). a , , / ( / ( a, + l) ) , . . . , / ( / ( a „ ) ) ) = Q(/(«

. . . , f (ai-1) , ai, f { ai + 1),...,f(a„)) = y as / is a translation and idempotent

on s é. Thus / leaves / (A) elementwise fixed.

Conversely, let / be a retraction on .vé. Consider the element У = Q( f ( a ! ) , . . . , f ( a i + 1),...,f(a„)). y = /( y ) , because y e f ( A )

and / (A) is a retract of s é . As / is a retraction on s é , y = f (y) = f { f (y)), and so y = / ( 6 ( / ( « i ) , f ( a i - i ) , at, f ( a i + l) , ..., f (a„))), Q ( f ( f ( a j ) , ...

. ■ . J ( f { a i„ f j ) , f { a i) J ( f { a i+1j ) , . . . J ( f { a n))) = ô ( / ( « i ) , / М = / (Ô te l, • • •, «n)), where/ ( / (a,)) = / (^) a s/ (a,) g/ (A) and/ {A) is a retract of s é , j = 1, . . . , « and j Ф i. Hence / is a translation on s é . Clearly / is idempotent. This completes the proof.

The proof above shows that for any translation Of on an algebra

sé = {A, J^>, Q e У*{A) is an ideal of s é .

As noted by Kolibiar [3], Theorem 1, each v-translation Я on a lattice

L induces an equivalence relation R À on L having the substitution property

over the operation v ; briefly JRA is a v -congruence on L. The next theorem gives a generalization.

Theorem 2. Let sé = <A, J ^ ) , Q e J * , be an algebra. There is a one-to-one

correspondence between idempotent translations 9* on sé and Q-congruences R on sé as follows:

(i) Any translation 9 determines a Q-congruence R c/ having property (iii)

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(ii) Any Q-congruence R having property (iii) determines a translation 9' R

as follows: STR(x) = x', where x R x ’ and x' belongs to the ideal U given in (iii).

(iii) There is an ideal U of sé such that each congruence class of

a Q-congruence R contains exactly one element of U.

P roof. Let ST be an idempotent translation on .я/. As shown above,

ST (A) is an ideal of

sé

. If C is a congruence class of Rv, ST (x) = ST (y)

for any two elements x, y of C, and so ST (A) n C contains one element only. Clearly each class C contains at least one element of ST (A). Thus it remains to show that Rf/ is a Q-congruence on

sT

. Rr is evidently an equivalence relation on

sé

according to its definition and so we consider the substitution property only. Let (x1}..., x„) and (yl5 ..., y„) be two u-tuples of A such that X;R</ y t for each value of i , i = 1 ,..., n. Then 9 ’ (Q(xl , ..., x„)) = У (У (Й (х 1,...,х„))) = y ( e ( x 1, ^ ( x 2) , . . . , ^ ( x j = е (.Г (х , | , ^ ( x 2) ,...

У (x„)) = Q(£f (yx), i f b i ) , у (y„)) = .... >’„)), whence R , has

the substitution property over Q.

Conversely, let R be a . Q-congruence on sT with the property given in

(iii), and let ST R be the mapping defined in (ii). We consider the element

Q (a i , . . . , an) e A. According to (iii), for any at there is an element u ^ U

such that üiRUi, i = 2 , . . . , n . Moreover, al Ral and Q(ax, u2, un)e U .

According to the definition of STR, STR(Q(al , ..., anj) = Q(at , u2, ..., u„)

= Q(a^, STR (a2) , ..., STR (anj). We have proved the Q-translation property for

the index 1, but the same construction of the proof holds for any indexvalue. Hence SfR is a translation on $2. As STR{a) and a belong to one and the same congruence class of R, STR [STR (a)) = STR(a).

The Q-congruence property allows us now to prove an analogue of Kolibiar’s Theorem 2 for idempotent translations on abstract algebras. Let

X and ST be two idempotent translations on s é . As usually, XST(a) — X[ST(a)),

and we say that X ^ ST if and only if R À ^ JR,/ .

Theorem 3. Let ST and X be two idempotent translations on an algebra

s2 — <A , J 5-) , Q e J*, and let X ^ ST. Then XST = STX = ST if and only if for each z for which 9 (z) = z also X(z) = z.

P ro o f. In other words, XST = STX = ST if and only if ST (A) ^ X(A). For any x e A , XST(x) = X(ST(xj) = ST(x), because ST(ST(x)) = ST(x). It follows from the definition X ^ ST о R A ^ R y that if X(a) = X(b), then

ST(a) = ST(b), because aRx b=>aR9 b. Let x e A be an arbitrary element.

Then А(Я(х)) = X(x) implies that ST(X(x)) — ST(x), whence STX(x) = ST (A(x)) = ST{x). Thus STX = XST = ST.

Conversely, let ST(z) = z. Then X{z) = X{ST(z)) = XST(z) = ST (z) = z, and

the desired property holds.

2. Ternary algebras and translations. An ordered pair sé = (A, Q) is called

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on . 9 such that for each a, b e A, Q(a, a, b) = a, and Q(a,b, c) is invariant under all six permutations of a , b , c e A . A ternary algebra is called normal, if for each four elements a, b, c, d e A, Q(Q(a, b, c), c, d) = Q(Q(a, d , c), c, b), and it is simple, if for each five elements a, b, c, d, ее A, Q(Q(a, b, c), d, e)

— Q(Q(a, d, e), Q(b, d, e), c). A simple ternary algebra is also normal [5].

The lattice structure of finite simple ternary algebras is characterized in [6], and the structure of algebras analogous to normal ternary algebras is studied under the title “Pseudoschar” by Hoehnke in papers [1] and [2].

Lemma 1. Each translation 9 on a ternary algebra s9

=

(A, Q) is idempotent.

P ro o f. a = Q ( a , a , b ) for each two elements a , b e A . Then 9 (a) = Q ( 9 (a), a, 9 (b)) and 9 ( 9 ( a ) ) = Q { 9 (а), 9 (a), 9 ( 9 (b))) = .9 {a).

Theorem 4. Let ,s9 — (A, Q) be a ternary algebra and 9 and X two

translations on s é . 9 and X permute, i.e. 9X(a) = X 9 (a) for each a e A , if and only if 9 (A) n X (A) # 0 .

P ro o f. Let d e X(A) n 9 ( A ) , and so 9 ( d ) = d — X(d), as 9 ( A ) and

X(A) are retracts of , 9. Now a = Q(d,a,a), and thus 9X(a) = 9(X(a)j = 9 (i. (Q(d, a,a))) = 9 { Q { l( d ), X(a), a)) = 9 ( Q ( d , Ma), a)) = Q(9(d),A(a), 9(a)) = Q(d,A(a),9(a)) = Q(A(d), A(a), 9(a)) = A(Q(d,a, 9(a))) = A{Q(9(d), a, 9(a))) = A{9{Q(d,a,a))) = A9(a).

Conversely, let X 9 (a) = 9 X (a) for any a e A and let b e 9 ( A ) . Then

X 9 (b) = X(b) = 9X(b), and so X ( b ) e 9 ( A ) . As X is idempotent, X(b)e X(A),

whence X(b)eX(A) n 9 ( A ) , and the theorem follows.

Now we can prove a uniqueness theorem analogous to [11], Corollary 2, and [14], Theorem 3.

Theorem 5. Let 9 and X be two translations on a ternary algebra

. 9 — (A, Q). X — 9 if and only if 9 ( A ) = X(A).

P roof. If 9 — X, then evidently 9 ( A ) = X(A). So, let 9 ( A ) = X(A). Now for each a e A, 9X(a) = 9(X(a)) = X(a) — X9(a) = X( 9 (a)) = 9 ( a ) , as 9 ( A ) = X(A) and 9X(a) = X9(a).

Let a, s e A be fixed elements of a ternary algebra s / = (A,Q). The set P(a,s) = {y| y = Q ( x ,a , s ) , x e A) is called the principal set generated by a and s in ..9.

Lemma 2. I f . 9 = (A , Q ) is a simple ternary algebra, then P(a,s) is an

ideal of . 9 .

P ro o f. We should show that when x , y e P ( a , s ) , then Q ( x , y , z ) , P(a,s) for any z e A . As x , y e P ( a , s ) , x = Q( r, a,s) and y = Q( t, a, s) for some elements r, t e A . Thus Q ( z , x , y ) = Q(z, Q(r, a, s), Q(t, a, s)) - Q( Q( z,r, t) ,

a, s) e P(a, s).

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Lemma 3. I f an ideal U of a normal ternary algebra sé is the retract of a translation 9 on s é , then U r\P(a, u) is a principal set of sé for any a e A and any u eU .

P roof. Let x eU n P(a, u). Then x = 9 (x) = ^ ( Q ( y , a, u)) = Q(9(y), <9* {a), и) and so x e P ( 9 (a), u). On the other hand, if x' e P( 9( a) , u), then x' = Q ( z , 9 ( a ) , u ) . Now Q ( Q ( z , 9 (a), u), a, u) = Q(Q(9(a), a, u), z, u), and

if 0 ( 9 (a), a,u) = 9 (a), then Q(Q (z, 9 (a), u), a, u) = Q (z, té (a), u) = x',

and so x ’ eP( a, u) . As 9 ( x ' ) = 9 ( Q ( z , 9(a),u)) = Q(z, 9 ( a ) , 9(u)) = Q(z, 9 (a), u) = x', x ' 6 U, and consequently, x ' e U n P(a, u). Hence, U n P ( a , u ) = P( 9( a) ,u) .

We must further show that Q(9(a), a, u) = 9 ( a ). As 9 ( a ) , u E U ,

Q(9(a), a, u)e U and so it is invariant under 9 . Thus Q ( 9 (a), a, u) = 9 (Q(£P(a), a, u)) = Q( ^( a) , £f(a), Sf {и)) = У (a). This completes the

proof.

Theorem 6. An ideal U of a simple ternary algebra sé = (A, Q) is a retract

of sé if and only if for any a e A and any u e U the set U n P ( a , u ) is a principal set of sé.

P roof. According to Lemma 3 it suffices to show the second part of the proof. Let U n P(a, u) = P(r, t). Let x e P ( r , t ) . Then x = Q{x, r, t ) and x = Q(a, u, x). Thus x = Q(Q(x, a, u), r, t) = Q(u, Q(a, r, t), Q(x, r, t)) = Q( x, u, Q(a,r,t)). Hence P(r,t) = P ( k , u ) for some k e U . We define the mapping 9* as follows: 9 ( a ) = k. Accordingly, 9 ( A ) <= U, and when a e U , then P(a,u) <= U, and so U n P(a,u) = P(a, u). Thus U Я 9 ( A ), and

9 ( A ) — U is an ideal of s é . As U n P(a, u) = P(a, u) for a e U , 9 leaves U elementwise fixed. We should further show that 9 is an endomorphism on s é .

According to the definition of 9 , we should show that U n P(Q(k, m, n), u) = Q ( u П P(k,u), U n P(m, u), U n P(n, и)) for any three elements k , m , n € A and u e U . We denote Q(U n P(k, u), U n P(m, u), U n P(n, u)) briefly by

Wkmn. Since u e U r \ P ( k , u ) , then U n P ( k , u ) is an ideal of sé (see [10],

Lemma 2). Moreover, Wkmn is an ideal of sé and Wkmn = (<2(x,y,z)|

z e U n P(k, u), y e U n P(m, u) and x e U n P ( n , u ) } as shown in [10],

Theorem 4. By results of Sholander (see [10]), we can transform sé into a meet-semilattice L, where the least element can be choosen freely,

Q(x, y, z) = (x л y) v (z л y) v (z л x), and the join of two elements x and y exists, if there is an element t > x, y in L. In our situation, we choose и as the least element. Let d e Wkmn. Then d = Q(x, y, z), where x e U n P(k,u),

and has the representation x — Q(x, к, и) = x л k; similarly у = Q(y,m, it) and z = Q(z,n,u). Now d = Q(x л к, у л m, z л n) = (x л к л у л m) v

v (х л к л z a n) v (у л т л z л п) ^ (к л m) v (к л n) v (т л n) = Q (к, т, п). Hence d — Q(d, Q(m, n, к), u) and thus d e U n P(Q(k, m, n), u).

Conversely, let b e U n P(Q(k, m, n), w); it means b = Q(b, Q(k, m, n), u)

= Q(Q(b, u, k), Q(b, u, m), n) = (b л k) v (b л m) v n. But Ь = Ь л Ь = (b л k) v

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v (b a m) v (b л n) = Q(Q(b, u, k), Q(b, u, m), Q(b, u,nj), where Q {b, u, k) e

e U n P ( k , u) as b, kg U and u, k e P ( k , u). Similarly Q(b, u, m)e U n P ( u , m) and Q ( b , u , n ) e U n P(n,u). Thus b e Wkmn and we can conclude that

Щтп = U о P(Q(k, m, n), u).

Let a, c e A be arbitrary elements of s é . We define the specified translation generated by a and c as follows: 9 ac(x) = Q( x,a, c). As the previous proofs suggests, the translation 9 ac characterizes simple ternary algebras.

Theorem 7. A ternary algebra sé = {A, Q) is simple if and only if

9 ac is a translation on , 9 for any two elements a, c e A.

P roof. Let 9 ac be a translation for any pair a, c e A. Then Q(Q(a, b, c), d, e) = ^ de Q(a,b,c) = Q ( 9 de(a), 9 de (b), c) = Q(Q (a, d, e), Q (b, d, e), c), and thus

, 9 is simple. The converse proof is similar.

A further problem on translations on sé is the characterization of direct sums of sé as done in [9].

References

[1] H.-J. H o e h n k e , Über Verallgemeinerungen des Scharbegriffs, Math. Nachr. 38 (1968), p. 365-382.

[2] — Zur Einbettung von Q-Pseudoscharen in Halbgruppen, Spisy prir. fak. Univ. J. E. Purkyne v Brne, 5 (1969), p. 131-142.

[3] M. K o lib ia r , Bemerkungen iiber Translationen der Verbdnde, Acta fac. r. Nat. Univ. Comenianae Math. 5 (1961), p. 455-458.

[4] L. N e b e sk ÿ , Algebraic properties of trees, Acta Univ. Carolinae Philologica Monographia XXV, Praha 1969.

[5] — Graphic algebras, Comment. Math. Univ. Carolinae 11 (1970), p. 553-544.

[6] J. N ie m in e n , Join-semilattices and simple graphic algebras, Math. Nachr. 77 (1977), p. 87-91.

[7] — A note on simple graphic algebras, Z. fUr math. Logik und Grundlagen der Math. 21 (1975), p. 365-367.

[8] — Derivations and translations on lattices, Acta Sci. Math. 38 (1976), p. 359-363. [9] — The lattice of translations on a lattice, ibidem 39 (1977), p. 109-113.

[10] — The ideal structure of simple graphic algebras, Colloq. Math. 40 (1978), p. 23-39. [11] M. P e tr ic h , On ideals of a semilattice, Czech. Math. J. 22 (1972), p. 361-367.

[12] J. P o es, Translations and endomorphisms in universal algebras, Acta fac. r. nat. Univ. Comenianae Math. 29 (1974), p. 95-99.

[13] G. S za sz, Die Translationen der Halbverbdnde, Acta Sci. Math. 17 (1956), p. 165-169. [14] — Translationen der Verbdnde, Acta fac. r. nat. Univ. Comenianae Math. 5 (1961),

p. 449^153.

[15] — and J. S z e n d r e i, Über die Translationen der Halbverbdnde, Acta Sci. Math. 18 (1957), p. 44-47.

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