CONTRIBUTIONS TO GENERAL ALGEBRA xx

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1 CONTRIBUTIONS TO GENERAL ALGEBRA xx Proceedings of the Klagenfurt Conference Verlag Johannes Heyn????, Klagenfurt ENDOMORPHISMS OF QUASIORDERS AND THEIR QUASIORDER LATTICE REINHARD PÖSCHEL AND SÁNDOR RADELECZKI Dedicated to the memory of Dietmar Schweigert Abstract. In this paper we describe binary relations σ A A preserved by the endomorphisms of a quasiorder q A A, i.e. with the property End q End σ. In particular we determine the quasiorder lattice of the algebra (A, Pol q) and conclude conditions for tolerance and congruence simplicity. We also characterize those quasiorders q A A whose endomorphism monoid End q is a dual atom in the poset ({End A A}, ). Introduction The quasiorders (i.e. compatible reflexive and transitive relations) of an algebra can be considered as a common generalization of its congruences and compatible partial orders. Their study has a particular importance in the case of partially ordered algebras, see e.g. [CzéL83b], [CzéL83a], [ChaPD99] or [RadS05]. For instance, any so-called order-congruence (see [CzéL83a]) of a partially ordered algebra (A, F) has the form q q 1, where q is a quasiorder of (A, F). The lattices of all quasisorders on a given set have several special properties (e.g. they are atomistic, dually atomistic and complemented, [ErnR95]). The situation changes if we consider quasiorder lattices of algebras: every algebraic lattice is isomorphic to the quasiorder lattice of a suitable algebra (see e.g. [ChaC96], [Pin95]). Only few properties of the lattice of quasiorders are known in the case of concrete classes of algebras (e.g. varieties). For majority algebras it is known that their quasiorder lattice is always distributive (see [CzéL83a] or [PinC93]). One of the starting points for this paper was the question how endomorphisms of quasiorders behave, in particular under which conditions we have Endq Endq for quasiorders q, q. This question is equivalent (see 1.1) to the question which q belong to the quasiorder lattice Quord(A, Endq). Therefore we investigate algebras (A, Pol q) ((A, End q), resp.) the operations of which are all the finitary (unary, resp.) operations preserving a given quasiorder q on an arbitrary set A. In particular we describe compatible binary relations σ A A of such algebras. As a corollary we get that the quasiorders of the algebra (A, Polq) form a particular (very small) distributive lattice (in general, (A, Polq) is not a majority algebra). We also prove that Endq is a dual atom in the poset ({End A A}, ) if and only if either q is a linear order or a nontrivial equivalence. This extents 2000 Mathematics Subject Classification. Primary: 08A35, 08A40, 08A02; Secondary: 06A99, 20M20, 08A60. Key words and phrases. Quasiorder of an algebra, endomorphism of a binary relation, clone, compatible binary relation. partially supported by DFG (436 UNG 113/173/0-1). supported by DFG-MTA Grant No. 167 and Hungarian National Research Found., Grant No. T and No. T

2 2 REINHARD PÖSCHEL AND SÁNDOR RADELECZKI some results of [GibZ98, Thm. 1] and [PecM05] dealing with the question when Pol Polσ or End Endσ for partial orders or equivalence relations, σ. We focus here mainly on quasiorders q but a detailed study of the proofs shows that several results can be generalized to reflexive relations q not necessarily transitive. 1. Preliminaries 1.1. Operations and relations. Let Op(A) (Op (1) (A), resp.) and Rel (2) (A) denote the set of all finitary (unary, resp.) operations and the set of all binary relations on a set A. The equivalence relations := {(a, a) a A} and := A A are called trivial. An n-ary operation f : A n A preserves a binary relation A A (or is invariant for or compatible with f) if (a 1, b 1 ),..., (a n, b n ) = (f(a 1,...,a n ), f(b 1,..., b n ) for all a 1,...,a n, b 1,...,b n A. We use the notation f or F if f preserves, or if every f F Op(A) preserves. For Rel (2) (A) and an algebra (A, F), where F Op(A), we introduce the notation Pol := {f Op(A) f }, End := {f Op (1) (A) f }, Inv (2) (A, F) := Inv (2) F := { Rel (2) (A) f F : f }, Con(A, F) := { Inv (2) (A, F) is an equivalence relation}, Tol(A, F) := { Inv (2) (A, F) is a tolerance relation}, Quord(A, F) := { Inv (2) (A, F) is a quasiorder} (see 1.2, 1.3). There are several Galois connections induced by, which were intensively studied (e.g. Pol Inv, Pol Inv (2), End Inv, for details see e.g. [PösK79], [Pös04]). Pol is always a clone, End is the endomorphism monoid of, Con(A, F) (Tol(A, F), resp.) is the congruence (tolerance, resp.) lattice of the algebra (A, F). Note that, for σ, Rel (2) (A), (End ) σ End Endσ σ Inv (2) (A, End ) Inv (2) (A, End ) Inv (2) (A, Endσ) Quasiorders. A quasiorder q on a set A is a reflexive and transitive relation q A A. The set of all quasiorders on A is denoted by Quord(A). For q Quord(A) the inverse q 1 := {(y, x) (x, y) q} is also a quasiorder and the relation q 0 := q q 1 is an equivalence on A. The q 0 -equivalence class of an element a A will be denoted by a/q 0 := {b A (a, b) q 0 }. Obviously, a quasiorder q is a partial order if and only if q 0 = q q 1 =. A binary relation A A is called connected 1 if ( 1 ) t = (here σ t denotes the transitive closure of a binary relation σ). Further, a partial order q is called locally bounded if any two elements a, b A have an upper bound and a lower bound in the poset (A, q). 1 i.e. the graph with vertex set A and edge set is connected in the usual graph theoretic sense

3 ENDOMORPHISMS OF QUASIORDERS... 3 Quord(A) forms a complete lattice with respect to inclusion. Meet and join are given by {qi i I} = {q i i I} and {q i i I} = ( {q i i I}) t for any q i Quord(A), i I, and least and greatest elements are and, respectively. The mapping q q 1 is an involution of the lattice (Quord(A),, ), whose fixed points q = q 1 are exactly the equivalence relations on A Quasiorders of algebras. The quasiorders q Quord(A, F) (= Quord(A) Inv (2) F, cf. 1.1) of an algebra (A, F) (i.e. the quasiorders compatible with F) form a complete sublattice of (Quord(A), ) (with meet and join as given in 1.2). The fixed points q = q 1 of the involution q q 1 are exactly the congruences of the algebra (A, F). Let P (1) (F) denote the set of all unary polynomials of the algebra (A, F) (i.e. the unary part of the clone generated by F and by all constant functions). It is well-known (see e.g. [RadS05, p. 44]) that (1.3.1) Quord(A, F) = Quord(A, P (1) (F)). For a quasiorder q Quord(A) (or more generally for any reflexive relation) we have P (1) (Polq) = Endq (since every constant mapping preserves q), thus the above equation implies in particular (1.3.2) Quord(A, Polq) = Quord(A, Endq). However, note that in general Inv (2) (A, Polq) Inv (2) (A, Endq) Endomorphisms. By definition, a mapping f : A A is an endomorphism of a relation Rel (2) (A) if and only if f, i.e. (f(x), f(y)) for all (x, y). We give some examples of special endomorphisms which we shall use later. (i) Let (a, a), (a, b), (b, a), (b, b) σ A A, a 0 A and let f : A A be defined by { b if x = a 0, (1.4.1) f(x) := a otherwise. Then f Endσ. Proof. Obviously, (f(x), f(y)) {(a, a), (a, b), (b, a), (b, b)} σ for arbitrary x, y A, in particular for (x, y) σ. (ii) Let q Quord(A) \ { }, (a, b) q, a b, b 0 A and let f : A A be defined by { b if (b 0, x) q, (1.4.2) f(x) := a otherwise. Then f Endq. Proof. Let (x, y) q. If f(x) = b then (b 0, x) q, and by transitivity of q we have (b 0, y) q, i.e. f(y) = b. Therefore (f(x), f(y)) {(a, a), (a, b), (b, b)} q. Thus f Endq. (iii) Let q Quord(A), ε := q q 1, and let A/ε = {X i i I} be the set of all equivalence classes of ε (the sets X i are just the connected components of q). For arbitrarily chosen c i X i and a mapping g : I I let f : A A be defined by (1.4.3) Then f Endq. f(x) := c g(i) if x X i (i I).

4 4 REINHARD PÖSCHEL AND SÁNDOR RADELECZKI Proof. Let (a, b) q. Then (a, b) ε = q q 1, i.e. i I : a, b X i. Consequently (f(a), f(b)) = (c g(i), c g(i) ) q, thus f Endq. 2. Main Result The main result of this section (Theorem 2.4) gives in particular a full description of the quasiorder lattice Quord(A, F) for algebras with F = Polq, where q is a quasiorder. Some of the consequences of the theorem (which are of its own interest, too) are given in the next section. As preparation for Theorem 2.4, the next lemmata give some conditions for binary relations compatible with the endomorphisms of a quasiorder q Lemma. Let q Quord(A) and σ Inv (2) (A, Endq). Then (i) σ \ q 1 = q σ, (ii) σ \ q = q 1 σ. Proof. Since (ii) follows from (i) (substitute q by q 1 ), it remains to prove (i). Choose (a 0, b 0 ) σ \ q 1 and let (a, b) q. For the mapping f : A A as defined in we have f(a 0 ) = a (since (b 0, a 0 ) / q because (a 0, b 0 ) / q 1 ) and f(b 0 ) = b (since (b 0, b 0 ) q). By 1.4(ii) we have f σ, thus (a, b) = (f(a 0 ), f(b 0 )) σ. This proves q σ Lemma. Let q Quord(A) and σ Inv (2) (A, Endq) \ { }. Then σ q 0 (= q q 1 ) = σ = q 0. Proof. By assumption there exists (a 0, b 0 ) σ with a 0 b 0. For any (a, b) q 0 q the function f (a,b) : A A defined by f (a,b) (x) := { b if x = b0 a otherwise is an endomorphism of q (according to 1.4(i)) and therefore preserves also σ. Hence we obtain (a, b) = ((f (a,b) (a 0 ), f (a,b) (b 0 )) σ, proving q 0 σ. Thus q 0 = σ Proposition. Let q Quord(A) and σ Inv (2) (A, Endq) \ { }. Then one of the following conditions is satisfied: (A) σ = q, (C) σ = q 0, (B) σ = q 1, (D) q q 1 σ 0 := σ σ 1. Proof. Clearly, one of the following four cases must appear: (a) σ \ q 1 and σ \ q =, (c) σ \ q 1 = and σ \ q =, (b) σ \ q 1 = and σ \ q, (d) σ \ q 1 and σ \ q. Case(a): By Lemma 2.1 we get q σ. As the assumption σ \ q = implies σ q, we obtain condition (A): σ = q. Case (b) yields (B): σ = q 1 (analogous to (a) substituting q by q 1 ). Case (c): this condition is equivalent to σ q q 1 = q 0. As σ, now Lemma 2.2 gives (C): σ = q 0. Case (d): By Lemma 2.1 we obtain q q 1 σ. Thus we also have q 1 q = (q q 1 ) 1 σ 1, hence (D): q q 1 σ σ 1 = σ 0. Now we are able to characterize the invariant relations of End q, in particular to describe the quasiorder lattices of the form Quord(A, Endq) for a quasiorder q. Note that these coincide with the quasiorder lattices of the (non-unary) algebras (A, Polq) (cf. 1.3) Theorem. Let q Quord(A). Then we have: (I) If σ Inv (2) (A, Endq), then one of the following two conditions is fulfilled: (i) σ {, q 0, q, q 1, q q 1, }, or (ii) q q 1 σ and, either σ q q 1 or \ (q q 1 ) σ. (II) Quord(A, Endq) = {, q 0, q, q 1, q q 1, }.

5 ENDOMORPHISMS OF QUASIORDERS... 5 Proof. (I): Let σ Inv (2) (A, Endq) \ { }. Then, according to Proposition 2.3, either σ {q 0, q, q 1 } (and we are done), or we have q q 1 σ 0 σ. In the latter case we consider ε := q q 1 = (q q 1 ) t Note that ε is an equivalence (since ε = ε 1 ). We are going to show σ ε or \ ε σ what will finish the proof. Thus assume σ ε. Then there must exist an (x 0, y 0 ) σ \ ε. Let X i1 := x 0 /ε, X i2 := y 0 /ε be the ε-equivalence classes of x 0 and y 0, respectively. Clearly, i 1 i 2, because (x 0, y 0 ) / ε. In order to show \ε σ, take any (x, y) \ε. Then there exist some j, k I such that x X j and y X k, where j k because (x, y) / ε. Now, choose elements c i X i, i I, such that c j = x, c k = y, and let g : I I be any function with g(i 1 ) = j, g(i 2 ) = k. According to 1.4(iii), the function f defined by belongs to End q. By assumption we have End q End σ, hence f σ and we get (x, y) = (c j, c k ) = (c g(i1), c g(i2)) = (f(x 0 ), f(y 0 )) σ (since (x 0, y 0 ) σ), proving \ ε σ and we are done. (II): The inclusion is clear since all indicated relations belong to the quasiorder lattice generated by q (cf. 1.2). The opposite inclusion follows from (I), since for a quasiorder σ the inclusion q q 1 σ implies ε := q q 1 = (q q 1 ) t σ t = σ. Therefore (cf. I(ii)), in case σ ε we have σ = ε, and in case \ ε σ, we get = ε ( \ ε) σ, thus σ = The lattice Quord(A, End q). According to Theorem 2.4(II) the quasiorder lattice (Quord(A, Endq), ) has at most six elements and its general form is given in Fig (a). Of course, in special cases some of the relations may collapse and one gets either a sublattice of the lattice Fig (a) or the lattice Fig (b) (see also 3.1). Hence the quasiorder lattice (Quord(A, Endq), ) is always distributive. q q 1 q q 1 q q 1 q 0 (a) (b) Figure The lattice (Quord(A, End q), ). 3. Applications In this section we collect some results which follow from Theorem 2.4 more or less easily Proposition. Let q Quord(A). The we have: (i) If q is a partial order on A then (cf. Fig (a)) Quord(A, End q) = {, q, q 1, q q 1, }. (ii) If q is a connected partial order on A then (cf. Fig (b)) Quord(A, End q) = {, q, q 1, }. (iii) If q is a linear order on A then Inv (2) (A, Endq) = {, q, q 1, }. (iv) q is an equivalence relation on A if and only if Inv (2) (A, Endq) = {, q, }.

6 6 REINHARD PÖSCHEL AND SÁNDOR RADELECZKI Proof. (i) and (ii) immediately follow from Theorem 2.4(II) and from the simple facts, that q 0 = for any partial order q, and q q 1 = (q q 1 ) t = for a connected partial order (see 1.2). If q is a linear order then q q 1 = q q 1 =, hence (iii) follows from Theorem 2.4(I). Finally, q is an equivalence if and only if q 0 = q = q 1 = q q 1 = q q 1, hence (iv) also follows from 2.4(I) Recall that an element a of a poset (P, ) with greatest element 1 P is called dual atom (or coatom) if it is covered by 1, i.e. a < 1 and a b < 1 implies a = b. Since Endq Endq 0 and Endq Endq q 1, by Theorem 2.4(II) we get that Endq (q Quord(A)) is a dual atom in the poset ({Endσ σ Quord(A)}, ) if and only if either q {q 0, q q 1 } what gives q 0 = q = q q 1, or q 0 = and q q 1 = (cf. Fig ), i.e. either q is an equivalence or a connected partial order (cf. 1.2). The next proposition will answer the more general question under which conditions the monoid Endq is a dual atom in the larger poset ({End Rel (2) (A)}, ). Clearly the greatest element of this poset is End = End = A A. For this we recall the following notions. A tolerance relation is a reflexive and symmetric binary relation. According to 1.1, Tol(A, F) denotes the set of all tolerances preserved by F. An algebra is called tolerance simple (congruence simple, resp.) if it has only trivial tolerances (congruences, resp.), i.e. if Tol(A, F) = {, }, (Con(A, F) = {, }, resp.) Proposition. Let q Quord(A). Then the following conditions are equivalent: (i) Endq is a dual atom in ({End Rel (2) (A)}, ). (ii) Either q is a nontrivial equivalence relation, or the algebra (A, End q) is tolerance simple and q / {, }. (iii) Either q is a nontrivial equivalence relation or q is a linear order. Proof. (i) = (ii): Let Endq be a dual atom (thus q / {, }, and let σ Tol(A, Endq) \ {, }, i.e. (A, Endq) is not tolerance simple. Then Endq Endσ A A, hence Endq = Endσ. With Proposition 2.3 we conclude that either σ {q 0, q, q 1 } or q q 1 σ. Assume by contradiction that q is not an equivalence, i.e. q is not symmetric. As now the cases σ = q and σ = q 1 can be excluded, let σ = q 0. Then Endq = Endq 0 and therefore q Inv (2) (A, Endq 0 ). Since q 0 is an equivalence, by Proposition 3.1(iv) we get q {,, q 0 } - a contradiction. Thus σ = q 0 is also excluded and therefore we must have q q 1 σ. Finally, as q is not symmetric, there exists (a, b) q such that (b, a) / q. Then (b, a) q 1 σ and (a, b) σ. According to 1.4(i), the mapping f(x) := { b if x = a a otherwise belongs to Endσ = Endq. But (f(a), f(b)) = (b, a) / q contradicts (a, b) q and f q. This shows that q must be an equivalence relation whenever (A, Endq) is not tolerance simple. (ii) = (iii): Let (A, Endq) be tolerance simple and q / {, }. Since q q 1 and q q 1 are tolerances of the algebra (A, Endq) and q q 1 q q q 1, we obtain q q 1 = (thus q is a partial order) and q q 1 =, what implies that q is a linear order. (iii) = (i): Suppose that Endq Endσ A A for some σ A A. Then σ Inv (2) (A, Endq) and σ / {, }, hence Proposition 3.1 (iii) and (iv) imply that σ {q, q 1 }. Thus Endσ = Endq = Endq 1, and this proves that Endq is a dual atom in ({End A A}, ). Finally we present some simplicity conditions which easily follow from our results. Some parts of them are implicitly contained also in [Sze86] and [KaaP01].

7 ENDOMORPHISMS OF QUASIORDERS Corollary. Let q Quord(A) \ {, }. (1) (A, Polq) (or equivalently (A, Endq)) is congruence simple if and only if q is a connected partial order. (2) (A, Polq) is tolerance simple if and only if q is a locally bounded partial order. (3) (A, Endq) is tolerance simple if and only if q is a linear order. Proof. (1) Let (A, Polq) be congruence simple. Since q q 1 and q q 1 are congruences of the algebra (A, Polq) and q q 1 q q q 1, we get q q 1 = and q q 1 =, thus q is a connected partial order. The if -part of (1) follows from 3.1(ii) and (2) Let (A, Polq) be tolerance simple. Then (A, Polq) is also congruence simple, and q is a partial order by (1). Since both, q q 1 and q 1 q, are tolerances of (A, Polq) and contain q, we get q q 1 = q 1 q =. These equalities imply that any two elements in the poset (A, q) have a common lower bound and an upper bound, i.e. q is a locally bounded partial order (cf. 1.2). Conversely, let q be a locally bounded partial order and let σ Tol(A, Polq) \ { }. We write now x y instead of (x, y) q. In [Sze86, Proof of 6.16, p. 151] was shown that then σ = what finishes the proof of tolerance simplicity. For completeness of this paper we recall this proof here (in a more general setting a similiar proof was given in [PösR06]). Thus fix (u, v) σ \ and choose (a, b) \ arbitrarily. Because q is locally bounded there exist upper and lower bounds c, d A, i.e. a c, b c, d a, d b. Consider the binary mapping f : A 2 A defined by c if u x, v y, (u, v) (x, y), or v x, u y, (v, u) (x, y), (3.4.1) f(x, y) = a if (x, y) = (u, v), b if (x, y) = (v, u), d otherwise. for x, y A. Then, as shown below, f preserves q. Consequently we have f Polσ (since Polq Polσ because σ Inv(A, Polq)). Since (u, v), (v, u) σ we get that (a, b) = (f(u, v), f(v, u)) σ. Thus σ = and we are done. It remains to prove f q. In fact, let x 1 x 2, y 1 y 2. We have to show f(x 1, y 1 ) f(x 2, y 2 ). According to we distinguish the following cases: Case 1: f(x 1, y 1 ) = d. Then f(x 1, y 1 ) f(x 2, y 2 ) because f(x 2, y 2 ) {a, b, c, d}. Case 2: f(x 1, y 1 ) = a. Then (x 1, y 1 ) = (u, v) and therefore u = x 1 x 2, v = x 2 y 2. Thus f(x 2, y 2 ) {a, c} what implies f(x 1, y 1 ) f(x 2, y 2 ). Case 3: f(x 1, y 1 ) = b analogously gives f(x 2, y 2 ) {b, c}. Case 4: f(x 1, y 1 ) = c. Then, either we have u x 1, v y 1 and (u, v) (x 1, y 1 ), or v x 1, u y 1 and (v, u) (x 1, y 1 ). The former case implies u x 2, v y 2 and (u, v) (x 2, y 2 ), therefore f(x 2, y 2 ) = c. The latter case analogously implies f(x 2, y 2 ) = c. Thus f(x 1, y 1 ) f(x 2, y 2 ). (3) If q is a linear order then (A, Endq) is tolerance simple because of 3.1(iii). Conversely, if (A, End q) is tolerance simple, then it is also congruence simple and the implication 3.3(ii) = (iii) shows that q must be a linear order Problems. The maximal clones (dual atoms in the clone lattice) on a finite set are known (see e.g. [Ros70]), among them are the clones Pol where is a bounded partial order or an equivalence relation on A. Since Polq Polq 0 Pol(q q 1 ) (this follows from e.g. 2.4), it is clear that in general Polq is not maximal for a quasiorder q. There are examples q where Pol q is a so-called submaximal clone (e.g. q = {(0, 1)} for A = {0, 1, 2}, see [Lau82] or [Lau06, Thm ]). We conclude with the problem to find all clones in the interval [Polq, Op(A)] (in the

8 8 REINHARD PÖSCHEL AND SÁNDOR RADELECZKI lattice of all clones) as well as to find all monoids in the interval [Endq, Op (1) (A)] (in the lattice of all submonoids of Op (1) (A)) Remarks. a) In Theorem 2.4 we explicitly describe Quord(A, F), but arbitrary invariant relations σ Inv (2) (A, F) are characterized only by special properties. Of course, there are known many such relations (e.g. q q 1 q q 1 and unions and intersections of these), but a complete explicite description is much more complicated and will be dealt with in another publication. b) We mention here the connection between quasiorders and closure operators and corresponding Alexandroff topologies (i.e. topologies where arbitrary intersections of open sets are open). Namely (see e.g. [Ern05] or [Ern91]), for a quasiorder q Quord(A), the operator ϕ : P(A) P(A) : B ϕ(b) is a closure operator (the so-called down-closure) where ϕ(b) := {a A b B : (a, b) q} is the down-set of B. Vice versa, for a closure operator ϕ we get a quasiorder via q := {(a, b) a ϕ({b})}. This establishes a correspondence between quasiorders and closure operators (which is injective on quasiorders). The corresponding (upper) Alexandroff topology T (A, q) is given by the closed sets of ϕ, i.e. the closed sets of T are the down-sets ϕ(b) of q (and the open sets are the up-sets ). The so-called specialization order q(t ) of a topology T, which in general is a quasiorder, is given by (x, y) q(t ) : O x O y (where O x denotes the set of all open sets containing x). Then q(t (A, q)) = q for any quasiorder q; moreover, the endomorphisms of q are exactly the mappings f : A A which are continuous with respect to T (A, q). Therefore, all our results can be easily transformed to corresponding results about Alexandroff topologies T on A and continuous mappings End(A, T ). Acknowledgments. This paper is a result of a collaboration of the authors within DFG-MTA Grant No This grant was originally planed and applied for by Dietmar Schweigert and the second author and confirmed by DFG. We gratefully acknowledge that after the unexpected death of Dietmar Schweigert in January 2006, the Deutsche Forschungsgemeinschaft (DFG) made it possible to start the project with the authors of this paper. Our thanks are also due to Marcel Erné who directed our attention to the connection to Alexandroff topologies and gave several valuable hints. References [ChaC96] I. Chajda and G. Czédli, Four notes on quasiorder lattices. Math. Slovaca 46(4), (1996), [ChaPD99] I. Chajda, A. Pinus, and A. Denisov, Lattices of quasiorders on universal algebras. Czech. Math. J. 49(2), (1999), [CzéL83a] G. Czédli and A. Lenkehegyi, On classes of ordered algebras and quasiorder distributivity. Acta Sci. Math. 46, (1983), [CzéL83b] G. Czédli and A. Lenkehegyi, On congruence n-distributivity of ordered algebras. Acta Math. Hung. 41, (1983), [Ern91] M. Erné, The ABC of order and topology. In: H. Herrlich and H.-E. Porst (Eds.), Category Theory at Work, Heldermann Verlag, Berlin, 1991, pp [Ern05] M. Erné, Minimal bases, ideal extensions, and basic dualities. Toplogy proceedings 29(2), (2005), [ErnR95] M. Erné and J. Reinhold, Intervals in lattices of quasiorders. Order 12(4), (1995), [GibZ98] P. Gibson and I. Zaguia, Endomorphism classes of ordered sets, graphs and lattices. Order 15(1), (1998), [KaaP01] K. Kaarli and A.F. Pixley, Polynomial completeness in algebraic systems. Boca Raton, Florida; Chapman & Hall/CRC., [Lau82] D. Lau, Submaximale Klassen von P 3. J. Inf. Process. Cybern. EIK 18(4/5), (1982), [Lau06] D. Lau, Function Algebras on Finite Sets. Springer Monographs in Mathematics, Springer, Berlin Heidelberg New York, 2006.

9 ENDOMORPHISMS OF QUASIORDERS... 9 [PecM05] M. Pech and D. Mašulović, On traces of maximal clones. Novi Sad. J. Math. 35(1), (2005), [Pin95] A.G. Pinus, On lattices of quasiorders on universal algebras. Algebra i Logika 34(3), (1995), , (in Russian; English transl. in Algebra Logic 34(1995), ). [PinC93] A.G. Pinus and I. Chajda, Quasiorders on universal algebras. Algebra i Logika 32(3), (1993), , (in Russian; English transl. in Algebra Logic 32(1993), ). [Pös04] R. Pöschel, Galois connections for operations and relations. In: K. Denecke, M. Erné, and S.L. Wismath (Eds.), Galois connections and applications, vol. 565 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2004, pp [PösK79] R. Pöschel and L.A. Kalužnin, Funktionen- und Relationenalgebren. Deutscher Verlag der Wissenschaften, Berlin, 1979, Birkhäuser Verlag Basel, Math. Reihe Bd. 67, [PösR06] R. Pöschel and S. Radeleczki, Endomorphisms of quasiorders and their quasiorder lattice. Preprint MATH-AL , TU Dresden, October [RadS05] S. Radeleczki and J. Szigeti, Linear orders on general algebras. Order 22(1), (2005), [Ros70] I.G. Rosenberg, Über die funktionale Vollständigkeit in den mehrwertigen Logiken. Rozpr. ČSAV Řada Mat. Přir. Věd., Praha 80(4), (1970), [Sze86] Á. Szendrei, Clones in universal algebra, vol. 99 of Séminaire de Mathématiques Supérieures. Les Presses de l Université de Montréal, Montréal, Reinhard Pöschel, Institut f. Algebra, TU Dresden, Dresden, Germany address: Reinhard.Poeschel@tu-dresden.de Sándor Radeleczki, Institute of Mathematics, University of Miskolc, 3515 Miskolc- Egyetemváros, Hungary address: matradi@gold.uni-miskolc.hu