DEMONSTRATIO MATHEMATICA Vol. XLI No 2 2008

Jerzy Ρ tonka

SUBDIRECT PRODUCT REPRESENTATIONS OF SOME UNARY EXTENSIONS OF

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 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 Subject Classification: Primary 06E05; Secondary 20N02, 08A05, 08B26. Key words and phrases: , 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]. Subdirect product representations 267

2. Subdirect decompositions Let us denote by Vo the variety of type To defined by (1.1), (1.2), (1.14) and (1.15). It is easy to check that LEMMA 2.1. The variety Vo is a subvariety of Vfc, k = 1,2,3. a Let an algebra 21 = • (A\ ·, / ) belong to 14, k = 1,2,3. We define in 21 2 2 two relation* s = and = as follows: for a, 6 € A we put a = b iff a = 6 ; we put a = b iff a = b or (a2 = a and 62 = b). LEMMA 2.2. The relation = is a congruence of 21 and (2l/¿) G Vo- Proof. Obviously = is an equivalence and it is a congruence of 21 by (1.8) and (1.9); see also [4], Section 5. The second statement holds by (1.10) and (1.11).

LEMMA 2.3. The relation = is a congruence of 21 and (2l/r*) € * Proof. Obviously = is an equivalence. By (1.12) the superposition law is satisfied for the dot operation. If a2 = a, then by (1.9) we have /a(o) = /α(α2) = (/a(a))2, so the superposition law holds also for The second statement holds by (1.12). •

r r* id id LEMMA 2.4. = C\ = = = , where = is the identity relation of 21 (defined by a = b iff a = b). THEOREM 2.5. An algebra 21 of type TQ belongs to Vk for k = 1, 2,3 if and only if 21 is isomorphic to a subdirect product of some algebras 2li and 2I2, where 21χ € Vo and 2l2 € V£. Proof. => Holds by Lemma 2.2, 2.3, 2.4 and Birkhoff 's subdirect decompo- sition theorem, see [2], p. 123. <= Holds by Lemma 2.1 and definition of Vk- •

3. Subdirectly irreducible algebras We shall consider the following 5 algebras of type TQ: 211 = ({αχ, 61}; -,/ai), where for every x, y G {αϊ, 61} we have

e ai x-y = < ^ and / (x) = χ. I αϊ, otherwise 212 = ({02,62}; 'j/212)) where for every x,y G {02,62} we have x-y = bi and f*2(x) = x. 213 = ({03,63}; •, f^3), where for every x,y (Ξ {03,63} we have x-y = 63 and f%3{x) = 63. 268 J. Plonka

2I4 = ({04,64,04}; -, Ζ2*4), where for every x,y G {04,64,04} we have 2t4 x-y = 64 and /^(o4) = c4, f**(a) = o4, / (64) = 64. a5 Sis = ({^5, 65, c5}; ·, / ), where for every x, y G {α5,65, c5} we have 5 x-y = 65 and /*(α5) = 65, /* (65) - 65, /«»(ce) = a5. Obviously, we have (3.i) iVone ίιυο of the algebras 2li, ..., 2I5 are isomorphic. In the sequel we shall use the following notation: If Β is a nonempty set we put 5(B) = {{χ} : χ £ B}. If a = (A; -,/a) is an algebra and Ρ is a partition of A inducing a congruence of 21, then this congruence we shall denote by ρ(Ρ). By Theorem 2.5 to find subdirectly irreducible algebras in I4 for k = 1,2,3 it is enough to find subdirectly irreducible algebras in Vo and V£. We have (3.ii) An algebra 21 of type tq is subdirectly irreducible and it belongs to Vo if and only if 21 is isomorphic to 21χ. Proof. Since Vo satisfies (1.14) and (1.15), so we can apply the standard and well known proof used for semilattices. • Let 21 = [A] ·, /a) be an algebra of type ro-

LEMMA 3.1. If 21 G and in the representation (l.iv) for a partition Π of A we have |/| ^ 3, then 21 is subdirectly reducible. Proof. Consider three partitions P\,P-¿, P3 of A for some different ¿1, ii-, ¿>3 S I, where

Pi = {Ah U Ai2} U S (A \ (Ah U Ai2)),

P2 = {Ah U Ai3} U S (A \ (Ah U Ah)),

P3 = {Ai2 U Ai3} U S (A \ (Ai2 U Ai3)). Then none of β(Ρ\), q(P2), q{P3) is the identity congruence of 21 but

ρ(Ρα) Π ρ(Ρ2) Π ρ(Ρ3) = i . LEMMA 3.2. If 21 G V* and in (l.iv) we have I = ¿2} with i\ φ and at least one of Ailt Ai2 is not 1 -element, then 21 is subdirectly reducible.

Proof. Put PI = {{ή,ΐ2}} U5(4\{¿i,¿2}), P2 = {Ah,AÍ2}. Then we argue as in the proof of Lemma 3.1. • LEMMA 3.3. If 21 G Vi and in (l.iv) we have I = {i), A — Ai is at least 3-element, then 21 is subdirectly reducible.

Proof. Let αι,θ2,ί be 3 distinct elements of A. Put Pj = {{aJ; ¿}} U \ {oj,î}) for j = 1,2. • Subdirect product representations 269

THEOREM 3.4. An algebra 21 is subdirectly irreducible and it belongs to Vj if and only if 21 is isomorphic to 2I2 or 2I3. Proof. <= Obviously, 2I2 and 2I3 are subdirectly irreducible, since they are 2-element. It is easy to verify that they satisfy (1.1)—(1.5). => By Lemmas 3.1-3.3, if 21 € V* and 21 is subdirectly irreducible, then it must be 2-element. Let a subdirectly irreducible algebra 21 = ({a, 6}; -,/a) belong to Vi- By (l.iv) we can assume that in 21 we have x-y = b, so by (l.iv) it must be /a(¿>) = b. By (l.iv) we have either /a(a) = a, so 21 is isomorphic to 2I2 or /a(a) = b, so 21 is isomorphic to 2I3. •

LEMMA 3.5. If 21 belongs to V2 and in (l.v) we have |Π| ^ 3, then 21 is subdirectly reducible. The proof is analogous to that of Lemma 3.1 but we consider involutions instead of stars.

THEOREM 3.6. An algebra 21 is subdirectly irreducible and it belongs to V2 if and only if 21 is isomorphic to 2I2 or 2I4. Proof. <= Follows from the fact that the congruences of 214 form a 3-element chain. => By Lemma 3.5, it must be |Π| < 3. It can not be |Π| = 1 since then by (l.v) A is 1-element, namely A = {ap0} and we do not consider 1-element algebras to be subdirectly irreducible. If |Π| = 2, then by (l.v) we have in 21 one loopoid 5ρ0 and one involution, so 21 must be isomorphic to 2I2 or 2I4. •

LEMMA 3.7. If 21 = [A] ·, /a) belongs to V3 and in (l.vi) we have \Ai\ > 1, then 21 is subdirectly reducible.

Proof. For j 6 A_put j = {¿} U {j} U (/^(j). Let ji,j2 £ A with

ή φ 32- Put Ρ: = {iJ U S(A \ ji), P2 = {j2} U S(A \ j2). LEMMA 3.8. If 21 belongs to V3 and in (l.vi) we have Ai = {j} and K/21) 1 (j ) j > 1, then 21 is subdirectly reducible.

2t -1 2l 1 Proof. PutP1 = {{z},{j},(/ ) (Í)}^2-{{¿,j}}u5((/ )- (j)). -

THEOREM 3.9. An algebra 21 is subdirectly irreducible and it belongs to V3 if and only if 21 is isomorphic to 2I3 or 2I5. Proof. <= Obvious. => Follows from Lemmas 3.7 and 3.8, since it must be \A{\ = 1 and MM e {0,1}.

We denote ir(Vi) = {2li,2l2,2l3}, fr(V2) = {2li,2l2,2l4}, Ir(V3) = {2l1,2l3,2l5}. 270 J. Plonka

Theorem 3.10. For every k = 1,2,3 we have: A subdirectly irreducible algebra 21 of type TQ belongs to Vk if and only if 21 is isomorphic to one of algebras from Ir(Vk). Proof. This follows from Theorem 2.5, (3.ii) and from Theorem 3.4 for k — 1, from Theorem 3.6 for k = 2 and from Theorem 3.9 for k = 3. • Theorem 3.11. For every k = 1,2,3 we have: An algebra 21 of type to belongs to Vk if and only if 21 is isomorphic to a subdirect product of some algebras from Ir(Vk), so Vk = HSP(lr(Vk)). Proof. This follows from Theorem 3.10 and Birkhoff's theorem on subdirect irreducibility. •

4. Minimal generics Lemma 4.1. For every i; = 1,2,3 toe have: An algebra 21 of type TQ is a generic of Vk if and only if 21 G Vk and Ir(Vk) Ç HSP(21). Proof. => Follows from Theorem 3.11.

<= It must be HSP(21) Ç Vk. Since Ir(Vk) C HSP(21), so by Theorem 3.11 we have Vk = HSP (Ir(Vk)) Ç HSP(21). - It was proved in [4], Section 5, Lemmas 5.1-5.3 that 21 (4.i) Let 21 = (A; -,/ ) G Vk, k = 1,2,3. Then the congruence classes of the congruence ξ can be indexed by the set I of idempotent elements ofQL, since in every congruence class [a] of = we have exactly one idempotent element i = a2. Moreover, every congruence class [a] of = is a subalgebra of 21 belonging to so it satisfies x-y = i. Lemma 4.2. For every k = 1,2,3 we have: If an algebra 21=(A; -,/21) is a generic of Vk, then > 3. Proof. If |/| > 3, then by (4.i) we have \A\ > 3. It can not be \I\ = 1, since then 21 E V£ and by (4.i) 21 satisfies (1.13). However (1.13) is not a consequence of regular identities defining each of Vk. If |/| = 3 and = 3 or |/| = 2 and \A\ = 2, then by (4.i) 21 satisfies (1.14) and (1.15), what is a contradiction. In fact, by Theorem 3.11 we have: If k = 1, then 2I3 €

Vi = HSP(21) and 2l3 does not satisfy (1.14) and (1.15). If k = 2, then by Theorem 3.11 we have 2I4 G V2 and if k = 3, then 2I5 G V3, so we get again a contradiction. Let finally, |/| = 2, I = {¿1,¿2} and a G [¿1], α φ i\. We show that there must exist a fourth element in A. In fact, if A = {¿1, a, ¿2}, then we have by (4.i) the following two possibilities: α (4.1) /«(α) =/ (ΐχ) = ¿ 1 and f\i2) = i2; Subdirect product representations 271

α a a (4.2) / (α) = α, / (»i) = ¿i and / (¿2) = i2. If (4.1) holds, then 21 satisfies (4.3) f(x) « x-x.

This however leads to a contradiction, since by Theorem 3.11 we have 2l2 G Vi, 2I4 G V2 and 2l5 G V3, where these algebras do not satisfy (4.3). If (4.2) holds then 21 satisfies: (4.4) f(x) « χ, what gives a contradiction again, since by Theorem 3.11 we have 2I3 G Vi,

SU € V2 and 2l5 G V3. • THEOREM 4.3. The subdirect product

ai 2 3 Sii,2,3 = ({(αϊ, ί>2, Ô3>, (αϊ, h, α3), (αϊ, α2,63), (h, 62, ¿>3)}; ·, / ' · )

0/ the 2li χ Sl2 x 2I3 is a minimal generic of Vi. Proof. It follows from Theorem 3.11, Lemma 4.1 and Lemma 4.2. •

(4.Ü) 2l2 G HSP (SU). In fact, put h(a4) = h(ci) = a2 and ^(64) = 62. Then h is a homomorphism. (4.iii) 2I3 G HSP (Sis). In fact, put fo(cs) = h(a5) = and h(b5) = 63. THEOREM 4.4. The subdirect product 21i,4= ({(01,04), (ai,C4>, (01,64), (61,64)}; -,/211'4)

0/ i/ie direct product 2li χ 2I4 is a minimal generic of V2. Proof. It follows from (4.ii), Theorem 3.11, Lemma 4.1 and Lemma 4.2. • THEOREM 4.5. The subdirect product 5 Sli,5= ({(ai,c5), (oi,o5), (01,65), (61,65)}; ·,Λ> ) of the direct product 2li χ 2I5 is a minimal generic of V3. Proof. It follows from (4.iii), Theorem 3.11, Lemma 4.1 and Lemma 4.2. •

Let s = (ρο,Ρι, · · ·, Pn, .. ·), η < ω, be a sequence of cardinals and let Ks be the class of all algebras 21 such that 21 represents the sequence s. We put g(s) = min {card (21) : 21 G /CS} if /CS is nonempty and g(s) = —1 otherwise.

THEOREM 4.6. g(S0) = 4 Proof. This follows from (l.ii) and Theorems 4.3-4.5. • We expect that the method used in this paper will be also useful in finding the numbers

References

[1] I. Chajda and K. Glazek, A Basic Course on General Algebra, Technical University Press, Zielona Gòra, Poland 2000. [2] G. Grätzer, Universal Algebra, 2nd ed., Springer-Verlag, New York-Heidelberg-Berlin 1979.

[3] G. Grätzer and A. Kisielewicz, A survey of some open problems on pn-sequences and free spectra of algebras and varieties, in: A. Romanowska and J. D. H. Smith (Eds.), "Universal Algebra and Quasigroup Theory", Helderman Verlag, Berlin 1992, 57-88. [4] A. W. Marczak and J. Plonka, Unarily extended semialttices, Asian-European J. Math., in print. [5] J. Plonka, On a method of construction of abstract algebras, Fundamenta Math. 61 (1967), 183-189.

MATHEMATICAL INSTITUTE OF THE POLISH ACADEMY OF SCIENCES ul. Kopernika 18 51-617 WROCLAW, POLAND E-mail: [email protected]

Received October 29, 2007.