Sauer, Norbert W.

Stone, Michael G.

English

Sauer Norbert W.

Stone Michael G.

University of Szeged

Acta Sei. Math., 38 (1977). 397—402 Endomorphism and snbalgebra structure; a concrete characterization N. SAUER and M. G. STONE § 1. Introduction In [5] the following abstract structure problem is solved: For what semigroups S and what lattices L does there exist an algebra 91 with S^ End 91, the endomor-phisms of 91, a n d L ^ S u 91 the lattice of subalgebras of 91? Here we provide a solu-tion to the corresponding concrete representation problem, where isomorphism is replaced by equality. Thus let SQAa be a given transformation monoid and LQ2A a set lattice. It is well known that Z,=Su 91 for some algebra 91 over the set A iff L is complete and compactly generated [2]; such lattices are called algebraic. In [4] necessary and sufficient conditions for S = E n d 91 for some algebra 91 over the set A are given; such transformation semigroups are called algebraic. A similar characterization is given in [4] for semigroups of partial functions. We make use of the latter result by representing subalgebras with partial identity functions to derive a simultaneous characterization for S and L. Our characterization, like that for the endomorphisms alone involves the solutions to systems of linear equations. If M is a set of partial functions on A to A with id € M the identity function on A, a system of linear equations Z over M is a set of f u n c t i o n a l equa t i ons each of the general form: fx=y, or fx=g with fig£M, together with a specified solution variable Xs. An assignment OL for I is a map from the variables of Z to partial func-tions on A to A with a common domain. The assignment a satisfies Z at d£A. prov ided f(ax(d))=oiy(d) whenever fx=y£Z a n d f(ax(d))=g(d) whenever fx=g£Z. The assignment a satisfies Z on DQ A iff a satisfies Z at d for each d£D. If X1 is the specified solution variable we say / is a solution to Z on D provided there is Received July 1, 1976. This research was supported in part by NRC Operating grants A7213 and A8094 from the Natio-nal Research Council of Canada. 13» 398 N . Sauer and M . G . Stone an assignment a which satisfies I on D and a(Xr)=f. A solution / t o E on D is unique provided f \D=h\D whenever h is any solution to E on D. The support of a system E is the set of all d£A for which there exists a solution to E at d. We write 5 = S p t E if B is the support of E. Denote by the algebra of all finitary operations which admit each / 6 M a a homomorphism. M is the set of all partial endomorphisms of 2IM and M is the set of all (total) functions which are endomorphisms of 3IM. As usual a partial function g is a homomorphism with respect to an operation P of rank v provided gP(x) is defined and equals P(gx) whenever gx is defined for x£Av. A total function is one whose domain is all of A. We will use: P r o p o s i t i o n 1. gÇAB belongs to M iff 2?€Su9I M and for each finite DQB there is a system E over M with g a unique solution to E on D. 13 r o o f. Take n = i n Theorem 2 of [4]. • § 2. The subalgebras of 9IM We first establish some easy facts about the support of systems E over M. L e m m a 1. If C = S p t E then there is an assignment a which satisfies E on C. P r o o f . For each d£C there is an assignment ccd which satisfies E a t d. Define a for a variable x of E by: f a d x ( d ) if d£C ax (d) — otherwise. It is straightforward to verify that a satisfies E on C. • L e m m a 2. If C = S p t I then there is a system F and an assignment p which satisfies r on C a n d C = S p t T and P(Xr)=id \ C is a unique solution to T on C. P r o o f . Let r have one additional new variable Xr not among those of E and let the equations of T consist of those of E together with the new equation X r = i d . By Lemma 1 there is an assignment a which satisfies E on C. Let P extend a by assigning id C to Xr. Clearly p satisfies r on C and C = S p t JT. If g is any solution to r on C then for dec, g(d)=d so g \ C=P(Xr)=id \ C thus id ï C is a unique solution to r on C. • L e m m a 3. Let each C^êF be the support of some system rc. Then f~) C is also the support of some system F. Endomo rph i sm and subalgebra st ructure; a concrete character izat ion 399 P r o o f . Assume without loss of generality that each pair of systems r c , F D have no variables in common for C^D and let Xr be a new variable distinct f rom all of those of the r c . By Lemma 2 we may further assume that id t C is a unique solution to r c on C for each Form r ~ ( (J r c ) U {Z r =id} . We claim Icejr ) D C = S p t r. If i / g S p t r , say a satisfies T at d, then clearly a c , the restriction of a to the variables of r c , satisfies f c at d for each J5" so d € H C. If on the other hand d£ H C and ac satisfies r c at d then let aX=acX \ p | C for a variable X in rc and let a X r = i d \ f | c Clearly a satisfies r on f | C so ¿£Spt T. Thus sptr= n c. L e m m a 4. For DQA the operation defined by D= H S p t I is a closure DgSpt2 . operator. P r o o f . Clevarly D g D, and [C Q D => C Q D]. To show D=D it is only necessary to see then that DQD. By Lemma 3 there is some system F with D= f l Spt 1 = _ DgSpti = S p t r . Clearly D= f ) SptI=Sptr=D. BE Spt2 L e m m a 5. For DQA the operation defined by D= ( J € is a closure Cfinite.Cgi) operator. P r o o f . Since d£D=>de{d}QD we have DQD. Further [CQD=>CQD] since each finite subset of C is also a finite subset of D. To show D=D it remains only to see D Q D . Suppose b % D \ then there is some a d D with a $ D . We will show this leads to a contradiction. Since a£D there is some BQD, B finite, with a£B. Thus f l Spt I, and say B— b„}. From BQD we have each Bgspt z _ n bK£D, say bK£CK for some CKQD, CK finite, K= 1, ..., n. Then C= | J CKis a finite K = 1 subset of D, so CQB. NOW BQ (J Q g (J CK so a£BQ (J CK = | J CK = CQD. __ K = 1 K = 1 _ K = 1 K = 1 Thus a£D, contrary to the original choice a({D, the desired contradiction. We can now describe explicitly the subalgebras of 2lM : T h e o r e m 1. 5GSu S&M iff B= U D. D finite, D£B P r o o f . Let B= 1J D. We first consider the case B=Q. Thus for each _Z> finite, DgjB DQB, D=0 and D = 0 so D = f | Spt I. If 9IM has any miliary operations 0gSpt2 (constants) a£A we claim a£Spz I for every system I. To see this let a be the as-signment which associates with every variable the constant function f :A --{a}. 400 N . Sauer and M . G . Stone Note a satisfies arbitrary I at a, since each g£M must have a as a fixed point. Thus from 5 = 0 we conclude 2iM has no nullary operations, whence B = 0 £ . €Su 9IM. Now if 1M0, fix an operation P of 9lM of rank n, and ax, a2,..., a„£B. It suffices to show that P(alt ...,an)€{au ...,an) since ...,an}^B. Let D = = {<*!, ..., a„}. If P(alt ...,an)$D then there is some Z with Z>Q Spt Z and P(au ..., a„)$Spt Z. By Lemma 1 we may assume that there is an assignment a which satisfies Z on Spt Z. We use a to produce an assignment a' which satisfies Z at d—P(px, ...,an) and thus obtain P(au ...,a„)£Spt Z contradicting the hypo-thesis that P(au ..., an)$D. For a variable x in £ let a'x be defined by a.'x(d\ = [ax{~d) \P(ux(ad,...,ax(aJ) if d = P(a1} ..., aB). We claim a' satisfies Z at d=P(au ..., a„). To see this consider an equation fx=y in Z: fr'x{d) = P(fctx(aj), ...Jax(aJ) = P{g(aJ,..., g(an)) = g j > ( f l l > . . . , a j = g(d) . Thus d£ Spt Z and we must conclude P{au ..., a„)£{a1,..., a„), so B= |J D=> D finite, D S B _ =*2?€Su 9lM. To prove the converse, we suppose B£Su 9IM. By Lemma 5 B<^B= = (J D so it remains only to show B^B. We proceed again by contradiction. D finite,DQB Let a£B and suppose aSince a£B there is some finite CQB with a£C. From the first part of the theorem we know C£Su 9lM. In fact C is the subalgebra of 9IM generated by C, for if Su 9IM and CQD then by the first part of our Theorem 1, D = (J G so CQD. Thus C is the smallest subalgebra of 2IM which contains C. G finite,GQD Now a£C so a=P(cx, ...,c„) for some operation P in 9IM and some sequence c 1 ; . . . , c„ from C. But 5 6 S u 9ÍM so B is closed under P, and C^B. Thus a=P(c x c„)£B. It follows that B^B and thus B£Su 5IM=>.B= |J D. • D finite, DQB § 3. Characterization Theorems For LQ2A let x(L)={f£AB\B£L, / = i d \ B} be the set o f characteristic func-tions of L. In general a function f£AB will be called a characteristic function if f=id[B. Recall (§1) that when M is a set of partial functions, M denotes the set of all partial endomorphisms of the algebra of all finitary operations which admit each f(LM as a partial endomorphism. In what follows LQ2A and M=SUx(L). T h e o r e m 2. <S=End 91 and Z , = S u 91 for some algebra 91 iff fd contains no total functions other than S and i t contains no characteristic functions other than x(L)-Endomorph i sm and subalgebra st ructure; a concrete character izat ion 4Ô1 P r o o f , Assume End 91 and L=Su 91 for some algebra 91. L e t / € ^ , /<f S. Since 5 '=End 91 some operation of 91 destroys / ; thisisame operation must admit each map in %(L) since Z = S u 91 so the operation is among those of 9IM. Thus the only total functions in M are the members of S. Now let g£AB with g = i d [ B, B$L, be a characteristic function. Then i?(£Su9I so there is some operation P in 91 and a finite sequence blt ..., b„£B with P(blt ..., b„)$B. This same operation P is again among the operations of 9IM . But P does not admit g as a partial endo-morphism since gP(bi, ..., b„) is undefined. Thus the only characteristic functions in i t are members of %(L). This proves one direction of the Theorem. To complete the proof let M= S U y_{L) and assume that M contains no total functions other than S and no characteristic functions other than x(L). Let 9 I=9I M . Since M is the set of all partial endomorphisms of 9IM we have 5=End9T . Moreover if 2?GSu9IM then id \ B is a partial endomorphism of 9IM so id \ B£ fit, thus id t B£-/_(L) so BeL. Thus £ = S u 9 I . • We now combine Theorem 2 with Proposition 1 and Theorem 1 to obtain an equational condition for S and L to be jointly algebraic. The characterization theorem which follows says roughly that S must contain all functions which are unique solu-tions to systems of equations over M=SU"/(L) and that the support of every such system must belong to L. (For A finite the theorem says exactly that ; the more general statement involves only additional "compactness" conditions which are "local" analogs of the above properties.) (1) and T h e o r e m 3. S = E n d 91 and L=Su 91 for some algebra 91 iff g€S. V finite DQA3 system I over M with g i D the unique solution to I on D (2) B = (J i f l s p t l ^ £ € £ . D f i n i t e D g B ( D E S p t Z . i o v e r M ) P r o o f . From Theorem 2 we know S = E n d 91 and L = S u 21 for some algebra 91 iff (i) g € A A and gÇJ&=>gÇS, and (ii) id\B£M => B^L. By Proposition 1 of § 1, (i) is equivalent to (1). Again by Proposition 1 of § 1, (ii) is equivalent to : (ii') 5 € S u 9 I M and V finite D Q B 3 system I over M with (id \ B)\ D the unique solution to I on D B£L. Furthermore the system I : {A' i=id} has id f D as a unique solution on each D , 402 N . Sauer and M . G . Stone thus (liO is equivalent to : [ f i g S u 3IM=>fl£Z,], and by Theorem 1 this is equivalent t o : (2) B = U i D )=>B£L. D finitcDSB v " = S p t r , I o v e r M / Thus 5 = E n d 21 and L=Su 21 iff (1) and (2) hold. • F o r A finite Theorem 3 can be restated simply and complete ly as: C o r o l l a r y 1. If A is finite and i d £ S Q A a and LQ2A then S = E n d 21 and L = S u 21 for some algebra 2t iff (1) g£ S whenever g is the unique solution to some system of equations with coefficients from SUx(L), and (2) B£L whenever B is the support of any system of equations with coefficients from S\Jx(L). Bibliography [1] M. ARMBRUST and J. SCHMIDT, Zum Cayleyschen Darstellungsatz, Math. Anrtalen, 154 (1964), 70—72. [2] G. BIRKHOFF and O. FRINK, Representations of lattices by sets, Trans. Amer. Math. Soc., 64 (1948), 299—316. [3] G. GRATZER, Universal Algebra. Van Nostrand (Princeton, 1968). [4] N. SAUER and M. G. STONE, The algebraic closure of a semigroup of functions (To appear). [5] M. G. STONE, Subalgebra and automorphism structure in Universal Algebras, a concrete characterization, Acta Sci. Math., 33 (1972), 45—48. DEPT. MATHEMATICS A N D STATISTICS UNIVERSITY OF CALGARY 2920 24 AVE. N.W. CALGARY, CANADA T2N 1N4 

Endomorphism and subalgebra structure; a concrete characterization

http://acta.bibl.u-szeged.hu/14649/1/math_039_fasc_003_004_397-402.pdf

Endomorphism and subalgebra structure; a concrete characterization

Abstract

Similar works

Full text

Available Versions

University of Szeged