1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References The endotopism semigroups of an equivalence relation Zhuchok Yurii, Toichkina Olena Luhansk Taras Shevchenko National University (Starobilsk, Ukraine) Berlin, October 13, 2017 Zhuchok Yurii, Toichkina Olena 1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References 1. Introduction Endomorphism semigroups of algebraic systems and their properties have been investigated by many authors (see, e.g., [1]{[4]). One of the first results related to endomorphisms of binary relations was obtained by Gluskin [5]; it states that any quasi-order relation is completely determined by the corresponding endomorphism semigroup. Later in this area of research numerous results for various classes of relations were obtained. For example, Shneperman [6] demonstrated that Gluskin's result cannot be carried over to the class of all reflexive binary relations, whereas in [7] this result was generalized to the class of so-called dense relations, and in [8], to a certain subclass of reflexive binary relations. Similar results for certain µ-ary relations were obtained by Popov [9], who introduced the notion of an endotopism. Zhuchok Yurii, Toichkina Olena 1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References 1. Introduction The notion of an endotopism is closely related to that of a correspondence, introduced by Kurosh [10] for an arbitrary universal algebra. As is known, the set of all endotopisms of a relation of any arity forms a semigroup with respect to the operation of componentwise multiplication. It turns out that the endotopism semigroup of any binary relation defined on some set is the correspondence of the symmetric semigroup on the same set. The various endotopism semigroups of an arbitrary equivalence relation are the main subject of study in the present talk. The regularity of the monoid of strong endomorphisms of finite undirected graphs without multiple edges was established in [11], Zhuchok Yurii, Toichkina Olena 1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References 1. Introduction finite n-uniform hypergraphs in [12], and infinite undirected graphs and hypergraphs in [13]. Regularity conditions for the endomorphism semigroups of equivalences were found in [14], for the endomorphism semigroups of ordered and quasi-ordered sets in [15] and [16], and for the endomorphism monoids of countable chains in [17]. An important subclass of regular semigroups is the class of coregular semigroups. The concept of coregularity on semigroups was introduced by Bijev and Todorov [18]. Investigations in this area were continued by Chvalina and Matouskova [19] and by Dimitrova and Koppitz [20]. Here we study conditions for regularity and coregularity of various endotopism semigroups for an equivalence. Zhuchok Yurii, Toichkina Olena 1. Introduction 2. Endotopisms of an equivalence relation 3. Endotypes of equivalence relations 4. Regularity and coregularity of correspondences 5. Representations of correspondences 6. References 1. Introduction Another concept considered in this work is the notion of the endotype. Depending on the conditions imposed on the endomorphism of a symmetric binary relation, Bottcher and Knauer [21] distinguished five types of endomorphisms, which were then used to define the endotype of the relation. Later, the notion of the endotype was defined for relations of any arity [22]. This concept can be used to classify relations by their endotype with respect to endomorphisms. For example, endotypes of generalized polygons were found in [23], those of the complement of finite paths in [24], and graphs of N-prisms in [25]. Here we generalize definitions introduced in [21] for endomorphisms to the case of endotopisms of binary relations and classify all equivalence relations by their endotype with respect to endotopisms. Zhuchok Yurii, Toichkina Olena 1. Introduction 2.1. Endotopisms and autotopisms 2. Endotopisms of an equivalence relation 2.2. Half-strong and locally strong endotopisms 3. Endotypes of equivalence relations 2.3. Strong and quasi-strong endotopisms 4. Regularity and coregularity of correspondences 2.4. The example of an endotopism 5. Representations of correspondences 2.5. Endotopisms of equivalence relations 6. References 2.6. Correspondences of End(α) 2.1. Endotopisms and autotopisms An ordered pair ('; ) of transformations ' and of a set X is called an endotopism [4] of ρ ⊆ X × X if the condition (x; y) 2 ρ implies (x'; y ) 2 ρ for any x; y 2 X . The set of all endotopisms of ρ equipped with the operation of componentwise multiplication forms a monoid denoted by Et (ρ). An ordered pair ('; ) of permutations ' and of a set X is called an autotopism of ρ ⊆ X × X if (x; y) 2 ρ if and only if (x'; y ) 2 ρ for any x; y 2 X . The set of all autotopisms of ρ endowed with the operation of componentwise multiplication forms a group denoted by At (ρ). Zhuchok Yurii, Toichkina Olena 1. Introduction 2.1. Endotopisms and autotopisms 2. Endotopisms of an equivalence relation 2.2. Half-strong and locally strong endotopisms 3. Endotypes of equivalence relations 2.3. Strong and quasi-strong endotopisms 4. Regularity and coregularity of correspondences 2.4. The example of an endotopism 5. Representations of correspondences 2.5. Endotopisms of equivalence relations 6. References 2.6. Correspondences of End(α) 2.2. Half-strong and locally strong endotopisms An endotopism ('; ) of ρ ⊆ X × X is called a half-strong endotopism if (x'; y ) 2 ρ implies the existence of x0 2 x''−1; y 0 2 y −1 such that (x0; y 0) 2 ρ. The set of all half-strong endotopisms of ρ is denoted by HEt(ρ). An endotopism ('; ) of ρ ⊆ X × X is called a locally strong endotopism if (x'; y ) 2 ρ implies that for any preimage x0 2 x''−1 there exists a preimage y 0 2 y −1 such that (x0; y 0) 2 ρ, and analogously for any y 0 2 y −1: The set of all locally strong endotopisms of ρ is denoted by LEt(ρ): Zhuchok Yurii, Toichkina Olena 1. Introduction 2.1. Endotopisms and autotopisms 2. Endotopisms of an equivalence relation 2.2. Half-strong and locally strong endotopisms 3. Endotypes of equivalence relations 2.3. Strong and quasi-strong endotopisms 4. Regularity and coregularity of correspondences 2.4. The example of an endotopism 5. Representations of correspondences 2.5. Endotopisms of equivalence relations 6. References 2.6. Correspondences of End(α) 2.3. Strong and quasi-strong endotopisms An endotopism ('; ) of ρ ⊆ X × X is called a strong endotopism if (x'; y ) 2 ρ implies (x; y) 2 ρ for any x; y 2 X . The set of all strong endotopisms of ρ endowed with the componentwise multiplication forms a monoid denoted by SEt (ρ). An endotopism ('; ) of ρ ⊆ X × X is called a quasi-strong endotopism if (x'; y ) 2 ρ implies that there exists a preimage x0 2 x''−1 which is adjacent with respect to ρ to every preimage from y −1; and analogously for suitable y 0 2 y −1: The set of all quasi-strong endotopisms of ρ is denoted by QEt(ρ): Zhuchok Yurii, Toichkina Olena 1. Introduction 2.1. Endotopisms and autotopisms 2. Endotopisms of an equivalence relation 2.2. Half-strong and locally strong endotopisms 3. Endotypes of equivalence relations 2.3. Strong and quasi-strong endotopisms 4. Regularity and coregularity of correspondences 2.4. The example of an endotopism 5. Representations of correspondences 2.5. Endotopisms of equivalence relations 6. References 2.6. Correspondences of End(α) 2.4. The example of an endotopism For any binary relation ρ on X we have a chain of inclusions: Et(ρ) ⊇ HEt(ρ) ⊇ LEt(ρ) ⊇ QEt(ρ) ⊇ SEt(ρ) ⊇ At(ρ): Let X = fa; b; c; dg, ρ = f(a; a); (a; c); (c; b); (d; a); (a; d); (d; d)g; then a b c d a b c d ; 2 SEt(ρ)nAt(ρ), a b c d d b c d Note that HEt(ρ); LEt(ρ) and QEt(ρ) do not form semigroups in general. If an endotopism ('; ) of ρ is such that ' = ; then we obtain the corresponding notion of an endomorphism. Zhuchok Yurii, Toichkina Olena 1. Introduction 2.1. Endotopisms and autotopisms 2. Endotopisms of an equivalence relation 2.2. Half-strong and locally strong endotopisms 3. Endotypes of equivalence relations 2.3. Strong and quasi-strong endotopisms 4. Regularity and coregularity of correspondences 2.4. The example of an endotopism 5. Representations of correspondences 2.5. Endotopisms of equivalence relations 6. References 2.6. Correspondences of End(α) 2.5. Endotopisms of equivalence relations Lemma. (i) A pair (τ; σ) of transformations (permutations) of X is an endotopism (autotopism) of α 2 Eq(X ) iff for any A 2 X /α there is B 2 X /α such that Aτ ⊆ B; Aσ ⊆ B (Aτ = B,Aσ = B). (ii) An endotopism (τ; σ) of α 2 Eq(X ) is a strong endotopism if and only if τ ∗ : X /α ! X /α : a 7! aτ is injective. (iii) An endotopism (τ; σ) of α 2 Eq(X ) is a quasi-strong endotopism if and only if (τ; σ) is a strong endotopism. (iv) An endotopism (τ; σ) of α 2 Eq(X ) is locally strong iff for all A 2 (X /α)τ ∗ and B; C 2 Aτ ∗−1 we have Bτ = Cτ,Bσ = Cσ: (v) An endotopism (τ; σ) of α 2 Eq(X ) is half-strong iff for any ∗ S A 2 (X /α)τ , (A \ X τ) × (A \ X σ) = Y 2Aτ ∗−1 (Y τ × Y σ): Zhuchok Yurii, Toichkina Olena Lemma.