DEMONSTRATIO MATHEMATICA Vol. XLI No 2 2008 Jerzy Ρ tonka SUBDIRECT PRODUCT REPRESENTATIONS OF SOME UNARY EXTENSIONS OF SEMILATTICES Abstract. An algebra 21 represents the sequence so = (0,3,1,1,...) if there are no constants in 21, there are exactly 3 distinct essentially unary polynomials in 2t and exactly 1 essentially n-ary polynomial in 21 for every η > 1. It was proved in [4] that an algebra 21 represents the sequence so if and only if it is clone equivalent to a generic of one of three varieties Vi, V2, V3, see Section 1 of [4]. Moreover, some representations of algebras from these varieties by means of semilattice ordered systems of algebras were given in [4], In this paper we give another, by subdirect products, representation of algebras from Vi, V2, V3. Moreover, we describe all subdirectly irreducible algebras from these varieties and we show that if an algebra 21 represents the sequence so, then it must be of cardinality at least 4. 1. Preliminaries By an algebra we mean a pair 21 = (A; Fα), where A is a nonempty set called the carrier of 21 and F21 is a set of finitary operations in 21 called the set of fundamental operations of 21. By the clone of 21 we mean as usually the smallest set containing all projections and closed under su- perpositions with fundamental operations of 21. We denote it by CJ (21). The operations from CI (21) are called polynomials, see [2], Two algebras 21 = (A\ F*) and 93 = (B; F®) we call clone equivalent if A = Β and a Ci (21) = Cl (05). A polynomial / (xi, ..., xn) depends on the variable xk, a k G {1, ..., η}, if there exist αϊ, ..., an, b G A such that / (ai, ..., an) φ α / (αι, ..., ak-1, b, ak+1, ..., α„). A polynomial 1, ..., xn) is essen- tially n-ary, if it depends on each of its variables. We denote by pn(21) the cardinality of the set of all essentially n-ary polynomials of 21, if η > 0. Next, we denote by po(2l) the cardinality of the set of all constant unary polynomials of 21, see [3]. 2000 Mathematics Subject Classification: Primary 06E05; Secondary 20N02, 08A05, 08B26. Key words and phrases: universal algebra, p„-sequence, generic algebra, semilattice ordered system of algebras, subdirectly irreducible algebra 264 J. Plonka Let ρ = (po, pi, ..., pn, ...), η < ω, be a sequence of cardinals. We say that this sequence is representable if there exists an algebra 21 such that PN(21) = Pn for all η = 0,1,2, Then we say that 21 represents the sequence p, see [3]. The problems and results connected with representable sequences were largely overlooked in [3] and they were considered by many authors. In the sequel we shall use some terminology and definitions from [1] and [2], Let το : {·,/} —{1,2} be a type of algebras, where το(·) = 2 and το(/) = I- We shall write x2 instead of x-x. We shall consider the following identities of type TQ: 1.1) x-y œ y-x 1.2) (x-y)-z « x-(y-z) 1.3) x2-y « x-y œ x-y2 1.4) f{x-y) « x-f(y) « f(x)-y « x-y 1.5) /(/(*)) « m 1.6) f(f{x)) « χ 1.7) f{f(x))*x2 1.8) (x-y)2 « x2-y2 1.9) (/Or))2 « f{x2) 1.10) (x2)2*x2 1.11) (/(s))2«*2 1.12) {x-y)2 « x-y 1.13) x-y u-v 1.14) x-x ÄS χ 1.15) f(x) ~ χ- It is easy to check that (l.i) The identities (1.8)—(1.12) are consequences of (1.1)—(1.4). Let 21 = (A; ·, /a) be an algebra of type tq and a G A. Then the element a will be called an f^-idempotent of 21 or briefly an f-idempotent if /a(a) = a; the element a will be called an idempotent of 21 if /a(a) = a = a • a. Subdirect product representations 265 Let Vi, V2, V3 be varieties of type to, where Vi is defined by (1.1)—(1.5), V2 is defined by (1.1)-(1.4) and (1.6), V3 is defined by (1.1)-(1.4) and (1.7). It was proved in [4] that (l.ii) An algebra 21 represents the sequence so = (0,3,1,1,.. •) if and only if 21 is clone equivalent to a generic of one of the varieties Vi, V2, V3. For k — 1,2,3 let V£ be the subvariety of Vfc satisfying identities of V& and additionally (1.13). In [4] it was proved that (l.iii) An algebra 2t of type TQ belongs to Vk if and, only if % is a semilattice ordered system of algebras from V£. For the definition of the sum of a semilattice ordered system of algebras we refer to [5]. In this paper we give another representations of algebras from Vk, k E {1,2,3}, namely by subdirect product (Section 2), what is convenient for finding subdirectly irreducible algebras from these varieties (Section 3). The structure of algebras from the varieties V£ for k E {1,2,3} is clear, since / yields a retraction in algebras from Vjf, / yields a permutation of order 2 in algebras from V£ and / yields a mapping of order at most 2, i.e., f^(x) ~ /I2'(a;) « χ2, in algebras from V3, cf. [4]. However, for considerations in Sections 3 and 4 we need more detailed description of algebras from V£, which we include below. For the convenience of the reader we use there a terminology taken from graph theory. An algebra £ = (A; / with one unary fundamental operation is called a unoid. A unoid $ = (A; / is called an instar if for some i E A and arbitrary χ E A the following holds: f^(x) = i. The element i is called the root of the instar J. For an algebra 21 = (A; ·, /a) of type TQ we denote by 21/ the unoid (A; /a) being the reduct of 21. (l.iv) Let 21 = (Α-, ·,/α) be an algebra of type r0. Then, 21 G Vf if and only if the following two conditions are satisfied: (1) there exists a partition Π = {^4¿}¿e/ of A such that for every i G I the set Ai is closed under f21 and the subunoid = [Al] f*\A.) is an instar; (2) for a fixed io Ε I and arbitrary x,y E A we have x-y = io, where ÌQ is the root of · Proof. => We denote by I the set of all /a-idempotents of 21. For i E I we put Ai = (/2t)_1(i). Since i is an /a-idempotent, it follows that i E Ai for every i E I. 266 J. Pîonka Let us prove (1). Since /a is a function, so if ¿1, ¿2 G I and ¿1 φ i<¿, then Π A12 = 0. By (1.5), we obtain the following: a a a a if a e A, then / (/ (a)) = / (a), so / (a) € I and a G Af*(a). If we put io = x-y for some x, y € A, then by (1.4) and (1.13) we get (2). This part of the proof is left to the reader. • A unoid S is called an involution, if it is of the form ({a, &}; where /®(a) = b and f^(b) = a. In particular, if a = b, then 5" is also called a loopoid. It means, that a loopoid is of the form ({a}; with /^(a) = a. (l.v) Let 21 = (A; ·, /a) be an algebra of type TQ. 21 G if and only if the following two conditions are satisfied: (1) there exists a partition Π of A such that for every Ρ G Π the set Ρ is closed under /a and the subunoid = (P;/a|p) is an involution; (2) for a fixed Po G Π and arbitrary χ, y G A we have x-y = ap0, where ap0 is the element of a fixed loopoid $p0. Proof. Put Π = {{a,b} : a,b G A,f*(a) = 6,/a(6) - a}. The rest of the proof is left to the reader. • A unoid J = (A] ft) is called an intree of height at most 2, if there is i E A and a partition Π = {{i}, Α^ U {Aj}jeA of A such that Í if ® e {¿} U Ai; \j, if χ G Aj, j G Ai. The element i is called the root of the intree (l.vi) Let 21 = (A; ·, /a) be an algebra of type TQ. Then 21 G V3 if and only if 21/ is an intree of height at most 2 and for x,y G A it holds x-y = i, where i is the root of the intree. Proof. For arbitrary α G A denote i = /a(/a(a)). Put Λ = (/a)(¿)\ and Aj = (/a)-1(i) for every j G Ai. The rest of the proof is left to the reader. • If 21 = (A; Fa) is an algebra, we denote card (21) = |A|. Let V be a variety of type r. Then an algebra 21 is called a generic of V if HSP (21) = V, see [2], Appendix 4. A generic 21 of V we call minimal, if for every generic 23 of V we have card (33) ^ card (21). If ψ rí ψ is an identity of type r, then it is called to be regular if the sets of variables in φ and φ are identical, see [5].