Generalized Quasiorders and the Galois Connection End-gQuord

Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) $\rho$ have the property that an $n$-ary operation $f$ preserves $\rho$, i.e., $f$ is a polymorphism of $\rho$, if and only if each translation (i.e., unary polynomial function obtained from $f$ by substituting constants) preserves $\rho$, i.e., it is an endomorphism of $\rho$. We introduce a wider class of relations -- called generalized quasiorders -- of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection End-gQuord, i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.


Introduction
Equivalence relations have the remarkable well-known property that an n-ary operation f preserves (i.e., f is a polymorphism of ) if and only if each translation, i.e., unary polynomial function obtained from f by substituting constants, preserves (i.e., is an endomorphism of ).Checking the proof one sees that symmetry is not necessary, thus the same property, called Ξ in this paper (see 2.2), also holds for quasiorders, i.e., reflexive and transitive relations.
No further relations with property Ξ were known and once we came up with the interesting (for us) question, if there are other relations (than quasiorders) which satisfy Ξ, we hoped to prove that Ξ( ) implies that has to be a quasiorder (or at least to be "constructible" from quasiorders).This attempt failed, but a new notion was born: transitivity of a relation with higher arity.The next step was to investigate reflexive and transitive m-ary relations which naturally are called generalized quasiorders for m ≥ 3 (for m = 2 they coincide with usual (binary) quasiorders) and which all have the property Ξ (Theorem 3.7).Moreover, these generalized quasiorders are more powerful than quasiorders or equivalence relations (see Remark 5.4) and therefore allow finer investigations of the structure of algebras (A, F ).
The next challenging question was: are there further relations with property Ξ, other than generalized quasiorders?The answer is "yes, but not really": there are relations satisfying Ξ( ) and not being a generalized quasiorder (see Example in 4.5), but each such relation is "constructively equivalent" to generalized quasiorders in the sense that they generate the same relational clone and therefore can be expressed mutually by primitive positive formulas (Proposition 4.4).
With the property Ξ the clone Pol of polymorphisms is completely determined by the endomorphism monoid M = End .Changing the point of view and starting with an arbitrary monoid M ≤ A A of unary mappings, one can ask for the set M * of all operations whose translations belong to M .Then Ξ( ) means Pol = (End ) * (for details see Section 2), in particular, M * is a clone.But in general, M * is only a so-called preclone (counterexample 2.4).This leads to the question When M * is a clone?and to the notion of a u-closed monoid (namely if M * is a clone).These u-closed monoids play a crucial role in this paper.Their characterization via generalized quasiorders, namely as Galois closed monoids (of the Galois connection End − gQuord introduced in Section 4), is one of the main results (Theorem 4.2) from which the answer to all above questions more or less follows.
The paper is organized as follows.All needed notions and notation are introduced in Section 1. Section 2 deals with the property Ξ and the u-closure and clarifies the preclone structure of M * .Section 3 is the stage for the main player of this paper: the generalized quasiorders.In particular, Theorem 3.7 proves the property Ξ for them.As already mentioned, in Section 4 the Galois connection End − gQuord and the crucial role of u-closed monoids is considered.Moreover, the behavior of the u-closure under taking products and substructures is clarified.In Section 5 we consider the u-closure of concrete monoids M ≤ A A , in particular all minimal u-closed monoids are determined (Theorem 5.3).In Section 6 we collect some facts and problems for further research.In particular we show how the notion of an affine complete algebra can be generalized via generalized quasiorders.

Preliminaries
In this section we introduce (or recall) all needed notions and notation together with some results.Throughout the paper, A is a finite, nonempty set.N := {0, 1, 2, . . .} (N + := N \ {0}) denotes the set of (positive) natural numbers.
1.1 Operations and Relations.Let Op (n) (A) and Rel (n) (A) denote the set of all n-ary operations f : A n → A and n-ary relations ⊆ A n , n ∈ N + , respectively.Further, let Op(A) = n∈N + Op (n) (A) and Rel(A) = n∈N + Rel (n) (A).
The so-called projections e n i ∈ Op (n) (A) are defined by e n i (x 1 , . . ., x n ) := x i (i ∈ {1, . . ., n}, n ∈ N + ).The identity mapping is denoted by id A (= e 1 1 ).C := {c a | a ∈ A} is the set of all constants, considered as unary operations given by c a (x) := a for a ∈ A.
Special sets of relations are Eq(A) ⊆ Quord(A) and Quord(A) ⊆ Rel (2) (A) of all equivalence relations (reflexive, symmetric and transitive) and quasiorder relations (reflexive and transitive), respectively, on the set A.

The Galois connection Pol
The Galois connection induced by gives rise to several operators as follows.
Preclones, also known as operads, can be thought as "clones where identification of variables is not allowed" (cf.1.4).The term preclone was introduced by Ésik and Weil [ ÉsiW2005] in a study of the syntactic properties of recognizable sets of trees.A general characterization of preclones as Galois closures via so-called matrix collections can be found in [Leh2010].The notion of operad originates from the work in algebraic topology by May [May1972] and Boardman and Vogt [BoaV1973].For general background and basic properties of operads, we refer the reader to the survey article by Markl [Mar2008].
Clones are special preclones.There are many (equivalent) definitions of a clone.One of these definitions is that a clone is a set F ⊆ Op(A) closed under 1.3(1)-( 6), [PösK1979, 1.1.2].Therefore we have: 1.4 Lemma.A preclone is a clone if and only if it is also closed under ∇ (adding ficticious variables) and ∆ (identification of variables).
For F ⊆ Op(A), the clone generated by F is denoted by F or F A .

The property Ξ and u-closed monoids
Equivalence relations or, more general, quasiorder relations have the remarkable property Ξ (see 2.2 below) that a polymorphism f ∈ Pol is determined by its translations trl(f ) defined as follows: 2.1 Definitions.For an n-ary operation f : A n → A, i ∈ {1, . . ., n} and a tuple a = (a 1 , . . ., a i−1 , a i+1 , . . ., a n ) ∈ A n−1 , let f a,i be the unary polynomial function  [Mal1963]) and let trl(f ) be the set of all such translations f a,i .For constants (as well as for arbitrary unary functions) f we put trl(f Given a set M ⊆ A A we define 2.2 Definition (The property Ξ).For a relation ∈ Rel(A) we consider the following property Ξ in three equivalent formulations: This can be extended to sets Q ⊆ Rel(A) just by substituting Q for in the above definition, e.g., Ξ(Q 2.3 Remark.As noticed above, it is well-known that Ξ( ) holds for ∈ Eq(A) or, more general, for ∈ Quord(A).Equivalently, expressed with the usual notions of congruence or quasiorder lattices, this means Con(A, F ) = Con(A, trl(F )) and Quord(A, F ) = Quord(A, trl(F )) for each algebra (A, F ) (F ⊆ Op(A)).
Clearly, there arises the question already mentioned in the introduction: Does there exist other relations with the property Ξ( )?
Since Ξ( ) implies that (End ) * is a clone and therefore it is closed under ∇ (cf.1.4).As we shall see in 2.5 below this also implies C ⊆ End , what expresses the fact that is reflexive (for definition see 3.2).However, the converse is not true: not each reflexive relation satisfies Ξ( ) as the following example shows.
Since M * is not always a clone, there also arises the question: what is the algebraic nature of the sets M * ?The answer gives the following proposition.4)).Thus we can consider the following two mappings between monoids and preclones:

Proposition. Let M ≤
ϕ : P → trl(P ), where P is a preclone on A.
ψ : M → M * , where M ≤ A A is a monoid on A.
Proof.Since, for a residuated pair (ϕ, ψ), the residual ψ is meet-preserving, the Lemma immediately follows from 2.6.We add a direct proof just using the definitions: Since M * is not always a clone, the question arises: For which monoids M ≤ A A the preclone M * is a clone?To attack this problem we introduce the u-closure M what shall lead to the equivalent problem (cf.2.9(iii)) of characterizing u-closed monoids.

Definition. For
2.9 Remarks.Let M ⊆ A A .
(i) The operator M → M is a closure operator (this follows from Lemma 2.7).
(ii) M is a monoid containing C and (M ) * is a clone (the latter follows from 2.7 because, by definition, M is the intersection of monoids N with N * being a clone; thus from 2.5 follows C ⊆ M , too).In particular we have follows from (ii), "⇐" follows from definition 2.8).
A characterization of u-closed monoids M will be given in the next sections (Proposition 3.9, Theorem 4.2 and Corollary 4.3).
3 Generalized quasiorders 3.1 Notation.Let A = {a 1 , . . ., a k } and M ≤ A A .We define the following |A|-ary relation: Thus Γ M consists of all "function tables" r g := (ga 1 , . . ., ga k ) (considered as elements (columns) of a relation) of the unary functions g in M .
In particular, we have Moreover, it is known that Pol Γ M coincides with the so-called stabilizer Sta(M ) of M and it is the largest element in the monoidal interval defined by M (all clones with unary part M form an interval in the clone lattice, called monoidal interval, cf., e.g., [Sze1986, Proposition 3

Definition (generalized quasiorder
).An m-ary relation ⊆ A m is called reflexive if (a, . . ., a) ∈ for all a ∈ A, and it is called (generalized) transitive if for every m × m-matrix (a ij ) ∈ A m×m we have: if every row and every column belongs to -for this property we write |= (a ij ) -then also the diagonal (a 11 , . . ., a mm ) belongs to , cf. Figure 1.
A reflexive and transitive m-ary relation is called generalized quasiorder.The set of all generalized quasiorders on the base set A is denoted by gQuord(A), and gQuord (m) (A) := Rel (m) (A) ∩ gQuord(A) will denote the m-ary generalized quasiorders.(i) Each quasiorder (i.e., binary reflexive and transitive relation) is also a generalized quasiorder.The converse is also true: Each binary generalized quasiorder is a usual quasiorder relation, i.e., we have gQuord (2) (A) = Quord(A).
Remark: Let A = {1, . . ., k}.We mention that for an n-ary function f : 3.4 Definitions.For ⊆ A m let tra denote the transitive closure of , i.e., tra = {σ ⊆ A m | σ is transitive and ⊆ σ} is the least transitive relation containing (it is easy to check that the intersection of transitive relations is again transitive).Analogously, the generalized quasiorder closure gqu is the least generalized quasiorder containing .The reflexive closure is naturally defined as These closures can be constructed (inductively) as follows.
3.8 Corollary.Let F ⊆ Op(A) and Q ⊆ gQuord(A).Then Proof.(i) directly follows from 3.7.Concerning (ii), we have Now we characterize the u-closed monoids M ≤ A A (i.e., M = M ) by various properties.The condition 3.9(iii) will show that the situation as in Example 2.4 is characteristic for being not u-closed.
3.9 Proposition.For a monoid M ≤ A A the following are equivalent: Proof.Each of the conditions (i), (ii) and (iv) implies C ⊆ M (cf.2.5 for (i), (ii) and note that Γ M is reflexive if and only if C ⊆ M ).Thus we can assume C ⊆ M in the following.Conversely, let f ∈ Pol Γ M , i.e., f Γ M .Remember that the elements of Γ M are of the form r g for some g ∈ M (notation see 3.1).Thus f Γ M means f (r g 1 , . . ., r gn ) ∈ Γ M whenever g 1 , . . ., g n ∈ M .Since f (r g 1 , . . ., r gn ) = r f [g 1 ,...,gn] , this equivalently can be expressed by the condition that the composition f [g 1 , . . ., g n ] belongs to M whenever g 1 , . . ., g n ∈ M .Consequently, any translation g := f a,i derived from f (w.l.o.g.we take i = 1), say g(x) := f (x, a 2 , . . ., a n ) for some a 2 , . . ., a n ∈ A, must belong to M , since g = f [id A , c a 2 , . . ., c an ] and M contains the identity id A and the constant functions.Thus trl(f ) ⊆ M , hence f ∈ M * , and we get Pol Γ M ⊆ M * .(iii) =⇒ (i): Assume (iii) and assume on the contrary that M * is not a clone.We lead this to a contradiction.Since M * is a preclone by 2.5, M * cannot be closed under ∆ and there must exist a function f ∈ M * , say n-ary, such that h := ∆f / ∈ M * (clearly n ≥ 3, otherwise we have a contradiction to (iii)).Thus some translation g := h a,i derived from h cannot belong to ∈ M (since g = ∆f by definition), in contradiction to (iii).
(iii) ⇐⇒ (iv): Let A = {1, . . ., k}.There is a bijection between binary operations f : A 2 → A and (k × k)-matrices (a ij ) via a ij = f (i, j) for i, j ∈ {1, . . ., k}.Note that rows and colums of (a ij ) are just the function tables (f (i, 1), . . ., f (i, k)) and (f (1, j), . . ., f (k, j)) of the translations f (i, x) and f (x, j).Therefore f ∈ M * (i.e., trl(f ) ⊆ M by definition) is equivalent to the property that all rows and colums of (a ij ) belong to Γ M (since the colums of Γ M are just the function tables of the unary functions in M ), i.e., Γ M |= (a ij ).Further, ∆f ∈ M is equivalent to the property that the diagonal (a 11 , . . ., a kk ) of (a ij ) belongs to Γ M .Thus condition (iii) is equivalent to the reflexivity (because C ⊆ M ) and transitivity of Γ M (according to 3.2), and therefore to Γ M being a generalized quasiorder.
The following corollary is a simple tool to construct functions in the u-closure of a monoid.

Corollary
Proof.The statement is just 3.9(iii) for the u-closed monoid M .
We mention further that h ∈ (M ) * is equivalent to Γ M |= V for the matrix V := (h(i, j)) i,j∈A .
4 The Galois connection End − gQuord 4.1 The Galois connection End − gQuord.The preservation property induces a Galois connection between unary mappings and generalized quasiorders given by the operators where the same Q can be taken in (ii) and (iii).
4.5 Remark.We know from 3.7 that ∈ gQuord(A) implies Ξ( ).The converse is not true: Ξ( ) does not imply ∈ gQuord(A) in general!A counterexample is the binary relation Before we investigate the u-closure for concrete monoids we show how this closure behaves under taking products and substructures.For this we need some notation.

Definition. Let g
Proof.(a): According to [PösK1979, 2.3.7] and because the identity map belongs to M i , for the invariant relations we have Inv . Thus, in order to prove (a), it only remains to show that for 1 ∈ Rel (m) (A 1 ) and 2 ∈ Rel (m) (A 2 ).But this follows from (notation see 3.2) what is clear from the definitions 4.6.
(b): Since the trivial equivalence relations ∆ A i and Satz 2.3.7(vi) and Üb 2.4, p.73] (restricting to unary mappings, i.e., taking End instead of Pol and Q i = gQuord M i ) in order to get the second equality in the following conclusions: Proof.According to [PösK1979, 2. σ ⊆ (since σ is transitive).If some row or column of (a ij ) contains an element a ∈ A \ B, then by definition of this row or column must be (a, . . ., a).Thus a ij = a for all i, j, and the diagonal obviously belongs to .Thus is transitive, i.e., ∈ gQuord(A).
"⊇": If ∈ gQuord(A) then σ := B ∈ Rel(B) is obviously reflexive (on B) and also transitive (since each matrix (b ij ) ∈ B m×m can be considered as a matrix in A m×m ).Thus σ ∈ gQuord(B).
4.9 Remark.We do not consider here the other side of the Galois connection, i.e., the Galois closures of the form gQuord End Q for Q ⊆ gQuord(A).

Minimal u-closed monoids
In this section we investigate some special monoids and their u-closure.For a unary function f ∈ A A let M f := f ∪ C.This is the least monoid containing f and all constants.What can be said about the u-closure of such monoids M f ?
In the following we have to deal much with the relation Γ M for a monoid M = M f and with the situation that Γ M |= V for some Therefore it is convenient to identify a g ∈ M with the vector r g = (ga 1 , . . ., ga k ) (cf. 3.1, here we assume A = {a 1 , . . ., a k } where A is implicitly ordered by the indices of a i ).Thus we can say that a row or column r of V equals some "vector" (k-tuple) g ∈ M and write r = g meaning r = (ga 1 , . . ., ga k ).This will be used very often in the proofs (in great detail in the proof of 5.1).Furthermore, let v i, * := (v i1 , . . ., v ik ) and v * ,i := (v 1i , . . ., v ki ) denote the i-th row and the i-th For the trivial monoid T := M id A = {id A } ∪ C we have: Proof.Let A = {a 1 , . . ., a k }.We show that Γ T is a generalized quasiorder (then we are done due to 3.9(iv)).Γ T is reflexive, thus it remains to show that Γ T is transitive.Let V = (v ij ) i,j∈{1,...,k} be a k × k-matrix such that Γ T |= V , i.e., each row and each column is one of the "vectors" g ∈ T , namely id A = (a 1 , . . ., a k ) or one of the constants c 1 = (a 1 , . . ., a 1 ),. . ., c k = (a k , . . ., a k ) (c i denotes the constant mapping c i (x) = a i ).If v jj = a i for some i = j, then Γ T |= V can hold only if all rows and columns are equal to the constant c i (since c i is the only vector where a i is on the j-th place), in particular, the main diagonal of V also equals c i and therefore belongs to Γ T .It remains the case v ii = a i for all i ∈ {1, . . ., k}.Then the diagonal of V is id A , also belonging to Γ T .Consequently, Γ T is transitive.
For |A| = 2 there exist only two monoids containing all constants, namely T and A A , both are u-closed (the first by 5.1, the second trivially).Therefore, in the following, we always can assume |A| ≥ 3.
We are going to characterize the minimal u-closed monoids, i.e., u-closed monoids M ≤ A A which properly contain no other u-closed monoid except the trivial monoid T = {id A } ∪ C. Such minimal u-closed monoids must be generated by a single function, i.e., they must be of the form M f for some unary f , moreover, M f can be assumed to be C-minimal, i.e., minimal among all monoids properly containing T (otherwise It is well-known which unary functions f generate a C-minimal monoid M f ≤ A A (it follows, e.g., from [PösK1979, 4.1.4]),namely if and only if f ∈ A A is a nontrivial (i.e., f / ∈ T ) function satisfying one of the following conditions: f is a permutation, such that f p = id A for some prime number p.
As shown in [JakPR2016, Theorem 3.1], among these functions are those for which the quasiorder lattice Quord f is maximal among all quasiorder lattices (on A), equivalently, for which End Quord f is minimal (among all endomorphism monoids of quasiorders).These functions are of so-called type I, II and III, defined as follows: (III) f is a permutation with at least two cycles of length p, such that f p = id A for some prime number p.
Note that f = {id A , f } for f of type I and II, and f = {id A , f, f 2 , . . ., f p−1 } is a cyclic group of prime order for f of type III.
Surprisingly it turns out (see Theorem 5.3) that for each candidate M f with f satisfying (i)-(iii), the u-closure M f is either not a minimal u-closed monoid or M f itself is already u-closed.Thus the minimal u-closed monoids coincide with the u-closed C-minimal monoids.We start with the functions of type I, II and III.
5.2 Proposition.Let f be a function of type I, II or III.Then M f is a minimal u-closed monoid, in particular M f = M f .Moreover we have End gQuord as it was explicitly stated in [JakPR2023, Theorem 2.1(B)] (but it already follows from the results in [Jak1982], [Jak1983] and also from [JakPR2018, Prop.4.8]).Thus we have equality instead of the above inclusions and M f is u-closed (by Theorem 4.2).Since M f has no proper submonoids except T because f satisfies one of the above conditions (i)-(iii), it is a minimal u-closed monoid.
II ) f 2 is a constant and |A| ≥ 4, or (III ) f p = id A for some prime p such that f has at least two fixed points or f is of type III.
In particular, each minimal u-closed monoid is C-minimal, too.
Proof.Part 1: At first we show that M f is u-closed for all functions of type I, and of the new type II or III .Because of Proposition 5.2, it remains to check only those functions which are of type II or III , but not of type II or III, respectively.
If v * ,i = f for some i ≥ 3, then all rows v , * must be equal to f for all ≥ 3 (in no other element of Γ M f appears 2 at the i-th place), consequently d V = (1, 1, 2, . . ., 2) ∈ Γ M f .The same arguments apply for the cases v i, * ∈ {f, c j } for some i ≥ 3 (change the role of rows and columns).
Thus it remains to consider the case that all v * ,i and v i, * (i ≥ 3) are neither f nor some c j .However then all these columns and rows were equal to id A , but this cannot appear because, e.g., v 3, * = id A and v * ,4 = id A would give v 34 = 4 and v 34 = 3, respectively, a contradiction.Note that here is used the fact k ≥ 4.
If v 2, * = id A , then we must have v * ,j = c j for j ≥ 3, thus d V = (1, 2, 3, . . ., k) ∈ Γ M f .If v 2, * = c 2 , then we must have v * ,1 = id A (recall v 11 = 1).Consequently, v j, * = c j for j ≥ 3 and we also get Case 2: f is of type III but not of type III, i.e., f p = id A for some prime p and the permutation f has only one cycle of length p but m fixed points z 1 , . . ., z m where m ≥ 2.
We have to show that Γ M f is transitive.Thus let Γ M f |= V where V is an (k × k)matrix V = (v ij ) i,j∈A (here we enumerate the rows and columns by the elements of A).
If v 00 = z is a fixed point z ∈ {z 1 , . . ., z m } then all columns and rows of V (as elements of Γ M f ) must be equal to c z , thus d V = (z, . . ., z) ∈ Γ M f .Let v 00 = i for some i ∈ Z p .Then v * ,0 ∈ {c i , f i }.
Assume v * ,0 = c i .If there exists some row v j, * = f i (for some j ∈ Z p ), then v j,z = z and therefore v * ,z = z for each z ∈ {z 1 , . . ., z m }.Thus the last m columns are all different, what implies v a, * = f i for all a ∈ A (here we need m ≥ 2).Consequently, Otherwise (if such a row v * ,j = f i does not exist), all rows v j, * must be equal to c i (j ∈ Z p ), what implies v * ,z = c i for z ∈ {z 1 , . . ., z m }, consequently d V = (i, . . ., i, . . ., i) ∈ Γ M f .The same arguments apply to the case v 0, * = c i resulting in d V ∈ Γ M f .Thus it remains to consider the case v * ,0 = f i and v 0, * = f i .However, this case cannot occur since then v z 1 , * = c z 1 and v * ,z 2 = c z 2 leads to the contradiction v z 1 ,z 2 = z 1 and v z 1 ,z 2 = z 2 (note m ≥ 2).
Part 2: Now we show that there are no more minimal u-closed monoids than those of type I, II and III .There are only the following two cases (A) and (B) for functions f to be considered for which M f is C-minimal (i.e., satisfies (i)-(iii)) but which are not of type I, II or III .We are going to show that for these f the u-closure M f is not minimal what will finish the proof of the Theorem.Case (A): f 2 is constant and |A| = 3.
There is only one (up to isomorphism) such function f on a 3-element set and we use the notation from Figure 4(A).Then M f = {id A , f, c 0 , c 1 , c 2 }.Consider the binary mapping h defined by the following table: Clearly h ∈ M * f (as indicated in the last column).Therefore (cf.3.10) g := ∆h ∈ M f where g (see Figure 4(b)) is a function of type I. Thus, by 5.2, we get M g = M g ⊂ M f , i.e., M f is not minimal u-closed.
Case (B): f p = id A , f consists of a single p-cycle and has at most one fixed point.
Consider the binary mapping h defined by the following table: Clearly h ∈ M * (indicated in the last column).Therefore (cf.3.10) g := ∆h ∈ M f and g is the permutation g : x → 2x for x ∈ Z p and gz = z.Note that 0 is an additional fixed point.First we consider the case that p ≥ 5.In the group generated by g there must exist an element g of prime order q with q < p.Since p ≥ 5, g has either more than one q-cycle or at least two fixed points, i.e., g is of type III .Since g ∈ g ⊆ M f we get (with 5.2) M g = M g ⊂ M f , i.e., M f is not minimal u-closed.
It remains to consider the cases p = 2 and p = 3.For p = 3, we get g = (0)(12)(z) (in cycle notation) if there exists a fixed point z what is a function of type III , and we can continue as above with g .Otherwise we have g = (0)(12).For p = 2 there must exist the fixed point z (since |A| ≥ 3) and we have f = (01)(z), what is a function of the same form as g in case p = 3 (up to isomorphism).Thus we can continue with g.Take the function h given by the table Then h ∈ M * g (as indicated in the last column) and therefore g := ∆h belongs to M g ⊆ M f .But g is a function of type I (g 0 = 0, g 2 = g 1 = 1).Thus, as above, M g = M g ⊂ M f , i.e., M f is not minimal u-closed.
5.4 Remark.Comparing Theorem 5.3 with the above mentioned results from [JakPR2016], we can conclude that there are monoids M ≤ A A which are characterizable by generalized quasiorders but not by quasiorders, i.e., we have M = End gQuord M but M End Quord M (namely those M f with f of type II or III but not of type II or III).With other words, generalized quasiorders are really more powerful than quasiorders (or congruences).
For |A| = 3, M. Behrisch (personal communication) computed all monoids of the form End Q for Q ⊆ gQuord(A) and of the form End Q for Q ⊆ Quord(A), their number is 89 and 71, respectively, among all 699 monoids M ≤ A A . ).In particular, M γ k is not u-closed (what was proved, at least for prime k = p, already with Part II, Case (B), in the proof of Theorem 5.3).

The lattices K (m)
A .For fixed A and fixed arity m ∈ N + , the set gQuord (m) (A, F ) of all m-ary generalized quasiorders of an algebra (A, F ) forms a lattice with respect to inclusion (where one can restrict F to unary mappings because of 3.7).All these lattices together also form a lattice, namely For m = 2 this lattice was investigated in [JakPR2016] (note Quord(A, F ) = gQuord (2) (A, F )). Due to the Galois connection End − gQuord the lattice K (m) A is dually isomorphic to the lattice of all those u-closed monoids M ≤ A A which are endomorphism monoids of m-ary generalized quasiorders.
The "largest" lattice K (k) A with k := |A| is isomorphic to the lattice of all uclosed monoids.With Theorem 5.3 we also determined the maximal elements of this lattice K (k) A , which are of the form gQuord M f with f satisfying one of the conditions I, II or III .

This K (k)
A contains all K (m) A for m < k via an order embedding.In fact, for m < n, there is an order embedding ϕ m n : A given by ϕ m n (gQuord (m) (A, F )) := gQuord (n) (A, F ) with F := End gQuord (m) (A, F ).
Conversely, there is a surjective order preserving map ψ n m : K A given by ψ n m (gQuord (n) (A, F )) := gQuord (m) (A, F ).This mapping is well-defined because gQuord (m) (A, F ) is "contained" in gQuord (n) (A, F )) since gQuord (m)  (it is easy to see that A n−m × is a generalized quasiorder if and only if is).Thus → A n−m × is an order embedding from gQuord (m) (A, F ) into gQuord (n) (A, F ). Instead of equivalence relations we may now consider other relations which also satisfy the property Ξ (cf.2.2).This leads to the notion generalized quasiorder complete, or gQuord-complete for short, which can be defined and characterized as follows:
Thus it is natural to ask which algebraic properties of affine complete algebras remain valid for gQuord-complete algebras.Moreover, what can be said about varieties generated by gQuord-complete algebras?
We recall that a variety V is called affine complete, if all algebras A ∈ V are affine complete.Similarly, we can define a gQuord-complete variety by the property that all its algebras A ∈ V are gQuord-complete.Hence, by our definition, gQuord-complete varieties can be considered a generalization of the affine complete varieties.It is known that any affine complete variety is congruence distributive (see e.g.[KaaM1997]).There arises the question what are the properties of gQuord-complete varieties, could they be still congruence distributive?In the paper [KaaM1997] also a characterization of affine complete arithmetical varieties is established (A variety is called arithmetical, if any algebra in it is congruence distributive and congruence permutable.)Therefore, it is meaningful to ask if there exists any characterization for gQuord-complete arithmetical algebras.
We mention some further topics for research: -Characterize the u-closed monoids which are already given by their quasiorders or congruences (cf.Remarks 5.4, 5.5), i.e., monoids M with the property M = End gQuord M = End Quord M or M = End gQuord M = End Con M .
-Investigate the lattices K

Remarks by two of the coauthors
In June 2022, a Honorary colloquium on the occasion of Reinhard Pöschel's 75th birthday was held in Dresden.There R. Pöschel presented a talk containing the basics of this article ([JakPR2022]).The colloquium was organized by M. Bodirsky and M. Schneider, who at the same time informed about a forthcoming topical collection of Algebra Universalis, which will be dedicated to R. Pöschel.At that time, the full version of the presented results was not yet written.
We, the co-authors of the results, somehow also would like to contribute to this honorary commemoration and therefore here -because we cannot submit it to the topical collection -we use the presentation of our common results as an opportunity to express our deep respect and gratitude to Reinhard, for his inventiveness, creativity, energy, and for his kindness.For more than 16 years we both have been working successfully together with Reinhard who was the initiator of many of our joint works.Our thanks also go to Martin Schneider for his activities.June 2023 Danica Jakubíková-Studenovská and Sándor Radeleczki

(
ii) =⇒ (i) =⇒ (iii) is clear (each set of the form Pol Q is a clone, and any clone is closed under ∆).
endomorphisms) and gQuord M := { ∈ gQuord(A) | ∀h ∈ M : h } (generalized quasiorders) for M ⊆ A A and Q ⊆ gQuord(A).The corresponding Galois closures are End gQuord M and gQuord End Q.Now we can show one of our main results, namely that the u-closed monoids are just the Galois closures with respect to the Galois connection End − gQuord.As a consequence (as shown in 4.3 and 4.4) we can answer the questions raised in the Introduction.4.2 Theorem.Let M ⊆ A A .Then we have: M = End gQuord M. Proof.At first we observe M ⊆ End gQuord M (this holds for every Galois connection), M ⊆ M = End Γ M , in particular M Γ M , and by 3.9(iv) we know Γ M ∈ gQuord(A).Thus Γ M ∈ gQuord M .Consequently we get M ⊆ End gQuord M = 3.8(ii) End gQuord M ⊆ End Γ M = M , and we are done.In addition to the characterization in 3.9 we give some further consequenses of Theorem 4.2, characterizing M * (4.3(a)) and u-closed monoids M (4.3(b)).Since every monoid can be given as endomorphism monoid of invariant relations, M = End Q, we also look for the characterization of those Q with u-closed endomorphism monoid (4.3(c)): 4.3 Corollary.(a) (M ) * = Pol gQuord M for M ⊆ A A .(b) The following are equivalent for M ≤ A A : Proof.(a): Let Q := gQuord M .Then Ξ(Q) by 3.7, i.e., Pol Q = (End Q) * (cf.(2.2.1)).Thus Pol Q = (End gQuord M ) * = (M ) * by 4.2.

Figure 3 :
Figure 3: The function f for Case 1 and Case 2 in the proof of 5.3

(
Remark 5.6) and their interrelations.Acknowledgement.The research of the first author was supported by the Slovak VEGA grant 1/0152/22.The research of the third author was carried out as part of the 2020-1.1.2-PIACI-KFI-2020-00165"ERPA" project -supported by the National Research Development and Innovation Fund of Hungary.