Rosenberg, Ivo G.

English

Rosenberg Ivo G.

University of Szeged

Acta Sci. Math., 43 (1981), 347—309 Functionally complete algebras in congruence distributive varieties I. G. ROSENBERG We say that QZ>Ah is central if ( a t , . . . , ah) € ¡? whenever a , = a j for some g i s invariant under permutations of coordinates and g ^ C x A h ~ 1 ^ 0 . ' We prove: A finite at least three-element algebra A = ( A ; F ) in a congruence distributive variety is functionally complete if and only if is simple, monotonic with respect to no bounded partial order on A and A1' admits no central subalgebra for H=2,..., \A\ — 1. For two-element algebras the condition simplifies to nonmonotonicity. If the variety K satisfies zf2(K), the absence of central subalgebras of Ah (h=2, ...,\A\-1) may be restricted to h=2. Recall that a finite algebra A is functionally complete (other names: complete or Sheffer with constants) if each finitary operation on the same universe is algebraic over A (i.e. obtainable from the operations of A, the projections and the constants via iterated composition). It is known (see e.g. [7; § 79] that all finite algebras in an arithmetical variety (i.e. in a congruence distributive and permutable equational class) are functionally complete. Recently McKenzie [5] has shown that with the exception of affine algebras a finite algebra in a congruence permutable variety is functionally complete if and only if it is simple (see also [2]; a short proof is in [15]). R. W. QUACKENBUSH in [7] asks for an analog of McKenzie's result for congruence distributive varieties. Combining JÓNSSON'S Mal'cev-type conditions [4] with the results of [8, 9] we answer this question. For a set A and h positive integer we say that gc:Ah is central if Q is totally reflexive ((alt ..., an)€g whenever a¡=a¡ for some 1 invariant under all transpositions of coordinates and contains CXAh~1¿¿0. The main result is: Theorem. A finite at least three-element algebra A=(A; F) in a congruence distributive variety is functionally complete if and only if A is simple, monotonic with respect to no bounded partial order on A and Ah admits no central subalgebra for h=2, ..., \A\ — l. Received November 10,1980. 348 I. G. Rosenberg Proof . The necessity is obvious. For example, if a nontrivial equivalence Q is a congruence of A then even the set of all operations on A admitting 9 as a con-gruence is not a functionally complete set of operations. Sufficiency. Let e-, ( /=1,2 ,3) denote the ternary projections (defined by ei(xi, x3)=xi for all Xi, x2, x3£A). It is well known [4] (quoted also in [1, Ch. 5, Ex. 70]) there exist and ternary operations e^tg, tx, ..., tn-±, tn = e3 derived from A such that for i=0,1, . . . ,« — 1 the identities (1) ti(x,y,x) = x, (2) ti(x, x, y) = r i+i(x, x, y) (i even), (3) ti(x, y, y) = ti+1(x, y, y) (i odd), hold for all x, yd A. To prove the functional completeness of A it suffices to verify that the operations of A augmented by the constants satisfy the following six types of conditions [8, 9, 11]. The first type of condition is just the nonmonotonicity. The conditions of the second type (the absence of the automorphisms of certain types) are taken care of by the constants. The third type of condition applies only if \A\=pm, p prime and requires A to be non affine. Here A is affine if all f£F are of the type (4) f ( X l , ...,x„) = B1x1+...+B„xn + C where + denotes the addition of an w-dimensional (column) vector space of char-acteristic p on A and Bt and C are mXm and m X l matrices over p = {0,1, ..., p — 1}. The following simple statement will be needed once more later. Claim 1. For at least one t£ T— {/l5 ..., /„} the following conditions (i)—(iii) are not equivalent: (i) t(x, x, y) = t(x, y, x) =x, (ii) t(x, y, y) = t(x, y, x) = x, (iii) t=e1. Proof . Suppose (i)—(iii) are equivalent for all if (/=1, ...,«). From t0=e1, (2) and (1) we obtain that tx satisfies (i), hence t1=e1 by (iii). From this, (3) and (1) it follows that t2 satisfies (ii) and again t2 = et. Continuing in this way we finally arrive at the contradiction e3=tn—e1. • Using (4) it is easily proved that the conditions (i)—(iii) are equivalent for t affine. Consequently, not all rf are affine and therefore A is not affine. The fourth condition is the simplicity of A. The fifth condition is that no h-ary central relation is a subalgebra of Ah for h—l, ..., \A\ — l. Our assumptions do not 349 I. G. Rosenberg: Functionally complete algebras in congruence distributive varieties cover the unary central relations (proper subsets of A) but these are taken care of by the constants. For the sixth type of condition, the first point to notice is that t0, ..., tn are all surjective (as maps A3-*A). We claim that at least one of the if is essentially at least binary. If not, then by (1) each t{ is either e± or e3. Let j be the least index such that tj = e3. Then clearly and using (2) or (3) (for z—x or z=y) we obtain the contradiction x = ex{x, z, y) = tj^{x, z, y) = tj(x, z, y) = y. Thus T—{tt, ...,?„} is not included in the set of all essentially unary or nonsurjec-tive operations which is a particular instance of the sixth type (the Slupecki con-dition [14]). We show that T satisfies the remaining conditions as well. To this end we must define( wreath algebras. Let h^3, and « > 1 be integers and let h={0, . . . , /z- l}, M-{i, ...,m},N={ 1, . . . ,«} and B=hm. A wreath operation on B is an n-ary operation w on B associated to permutations pt of b (¿=1, ..., m) and. maps r: M-~N and s: M—M as follows. For xi = (xa, ..., xim)£B (i£N) set (5) *>(*!, ..., XN) = (Pi(x,(i)s(i)), PM (-*j*(m)s(m))) • Now an algebra A is said to be a wreath algebra if it is isomorphic to an algebra, on B having wreath operations only. Next we prove the following: Claim 2. For a ternary wreath operation t on B the conditions (i)—(iii) above are equivalent. Proof . Let py, p2, p3 be the permutations of h and let r: M—{ 1, 2, 3} and s: M—M be the maps in the representation (5) of t. Set Ri = r~1 {¡} (/=1, 2, 3). To prove (i)^(iii) observe that in (5) we need R2—R3=0', moreover s and pt ( f= l , 2, 3) must be identities on M and h. The same argument proves (ii)=>(iii). Finally (iii)=>(i) and (iii)=>(ii) are obvious. • Using an idea from [13] (applied also in [10]) it was shown in [12] that a surjec-tive algebra does not satisfy the remaining conditions of the sixth type if and only if it is a wreath algebra. Combining both claims we obtain that (A, T) cannot be a wreath algebra; consequently A is not a wreath algebra and the proof is com-plete. • Remark 1. For algebras on a two-element set the situation is much simpler.. There are only two bounded partial orders (dual to each other) on such a set and therefore a single type of monotonicity. Similarly there is but one type of linearity. It is well known and follows from Post's criterion [6, 3] that a two-element algebra, is functionally complete if and only if it is neither monotonic nor linear. The same. 350 I. G. Rosenberg argument as above may be used to remove the nonlinearity stipulation yielding: A two-element algebra in a congruence distributive variety is functionally complete if and only if it is not monotonic. Remark 2. Observe that there are more exceptional cases than in congruence permutable varieties; moreover they are of a different nature. Lattices — a typical instance of congruence distributive varieties — provide examples of functionally noncomplete algebras that are monotonic and possibly not simple. For a congruence distributive variety K let /d„(K) denote the existence of ex= = t0, tx, ..., t„—e3 satisfying (1)—(3) for the least n [4]. For example, <d2(K) means that each algebra in K has a majority ternary operation M among its derived opera-tions (i.e. M satisfies the identity M(x, x, y)=M(x, y, x)=M(y, x, x)=x). We have: Corol lary. Let K be an equational class of algebras satisfying zl2(K). A finite algebra A in K is functionally complete if and only if it is simple, monotonic with respect to no bounded partial order on A and admits no central subalgebra of A2. Proof . Let 2</z<| / l | and let Q be central. Then there exists c f A such that cXAh~1Qe- Let M be the corresponding majority operation. We maintain that e is not a subalgebra of (A; M)h. Assume it is and choose alt ..., ah£A. From M(aj_, QY, c) = ax, M(a2, c, a2) = a2, M(c, a¡, a¡(i=3, ..., h) and {ai, a2, c, ..., C)£Q, («I, c, a3, ..., ah)£g, (c, a2, ..., ah)dQ it follows that (ax, ..., ah)£g leading to the contradiction Q=AH. • Remark 3. Applying an argument from [12] it can be shown that for a sur-jective algebra (i.e. the operations are all onto maps) in a congruence distributive variety the absence of central subalgebras can be restricted to the nonexistence of binary central subalgebras. Now for « > 2 and we construct a functionally noncomplete algebra A¡ in a variety K satisfying /1„(K) having central h-ary subalgebras exactly for h in the range from / to \A\ — 1 and satisfying all the other conditions of the theorem. For this end we first exhibit a ternary algebra TC=(A; t0, ...,?„} satisfying (1)—(3) and admitting every central /r-ary relation Q containing {c}XAh~1 (h=2, ...,\A\ — \) where c is a fixed element of A. Set (6) t^x, x, y) = x, ti(x,y,x) = x (i = 1, . . . , n - l ) , (7) t„-i(,x, x,y) = y (n odd), í„_i(*, y, y) = y (n even) and t¡(x, y, z)=c in all remaining cases. To establish that a central relation Q containing {c}xAh~1 is a subalgebra of Thc it suffices to prove the following claim. If 1 = 2 a = (a1, ..., ah)£Ah\Q and x2J, x3j) = a ; (j=\, ..., h), then a—(xml, ...,xmh) for at least one Note that all a¡ are distinct from c Functionally complete algebras in congruence distributive varieties 351 because For / = 1 by virtue of (6) then either xlj=x2j=aj or x1J=x3j=aJ and therefore a=(xu, ..., xlh). Similarly by (7) we obtain a=(x31, ..., x3h) for i=n—1 while a = ( x u , ..., xlh) follows directly from (6) for —1. Let U[ denote the set of all finitary operations on A whose range has less than / elements (i.e. |/(,4")|</). Finally At = (A; C/,U{?0, ..., i„}> provides the required example. Indeed for l^h<\A\ due to total reflexivity each central /i-ary relation Q is a subalgebra of (A;f)h for every f£Ut whereas for 1 <&</ there is always a range h operation / not admitting g as a subalgebra of (A; f ) h . To conclude we mention the following problem arising in this connection. Suppose A is a finite algebra in a congruence distributive variety which is not func-tionally complete (e.g. a lattice). Functional completeness is achieved by adjoing new operations. The problem is to describe conditions for these added operations. These will depend on the conditions of the theorem failed by A and a fortiori by <0,..., tn which may impose certain structure on t0, ..., tn (for example the mono-tonicity of all t0, ..., t„ with respect to a bounded partial order) and allow us to restrict the conditions for new operations. References [1] G . GRÄTZER, Universal Algebra, Van Nostrand (Princeton, 1968). [2] H. P. GUMM, Algebras in permutable varieties: Geometrical properties of affine algebras, Algebra Universalis, 9 (1979), 8—34. [3] S. V. JABLONSKII, G . P. GAVRILOV and V. B . KUDRJAVCEV, Functions of the Algebra of Logic and Post Classes (Russian), Izdat. Nauka (Moscow, 1966), 119 pp. German translation Boolesche Funktionen und Postsche Klassen. Akademie Verlag (Berlin, 1970). MR35# 6489, 43 #2868 14] B . JÖNSSON, Algebras whose congruence lattices are distributive, Math. Scan., 2 1 ( 1 9 6 7 ) , 110—121. [5] R. MCKENZIE, On minimal locally finite varieties with permuting congruence relations, pre-print (1976). [6] E. L. POST, Introduction to a general theory of elementary propositions, Amer. J. Math., 43 (1921), 163—185. [7] R . W . QUACKENBUSH, Primality: The Influence of Boolean Algebras in Universal Algebra. Appendix to the second expanded version of G. Grätzer's Universal Algebra, Springer Verlag (New York, Heidelberg, Berlin, 1979), 401—416. [8] I. G. ROSENBERG, La structure des fonctions de plusieurs variables sur un ensemble fini. C. R. Acad. Sei. Paris, Ser. A—B, 260 (1965), 3817—3819. MR31 #1185 [9] I. G. ROSENBERG, Über die funktionale Vollständigkeit in den mehrwertigen Logiken, Roz-pravy Ceskoslovenske Akad. Vid., Hada Mat. Pfir. Ved., 80:4 (1970), 1—93. MR45#1732 [10] I . G . ROSENBERG, Complete sets for finite algebras, Math. Nachr., 4 4 ( 1 9 7 0 ) , 2 2 5 — 2 5 8 . MR43#4639 [11] I.G.ROSENBERG, Completeness properties of multiple-valued logic algebras, in Computer Science and Multiple-Valued Logic, Theory and Applications (Ed. D. C. Rine), North-Hol-land (1977), Ch. 6, 144—186. 8 352 I. G. Rosenberg: Functionally complete algebras in congruence distributive varieties [12] I. G. ROSENBERG, Functional completeness of single generated or surjective algebras, in Finite Algebra undMultiple-Valued Logic, Coll. Math. Soc. J. Bolyai, Vol. 28, North-Holland (Amsterdam, 1981), 635—652. [13] G. ROUSSEAU, Completeness in finite algebras with a single operation, Proe. Amer. Math. Soc., 18 (1967) 1009—1013. MR36#97 [14] J. SLUPECKI, Completeness criterion for systems of many-valued prepositional calculus, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Cl. II, 32(1939), 102—109 (Polish). English translation: Studia Logica, 30 (1972), 153—157. MR47# 8245 [15] Â. SZENDREI, A new proof of the McKenzie—Gumm theorem, Algebra Universalis, to appear. UNIVERSITÉ D E MONTRÉAL CASE POSTALE 6128, SUCC „A" MONTRÉAL H3C 3J7 

Functionally complete algebras in congruence distributive varieties

http://acta.bibl.u-szeged.hu/14824/1/math_043_fasc_003_004_347-352.pdf

Functionally complete algebras in congruence distributive varieties

Abstract

Similar works

Full text

Available Versions

University of Szeged