Complete Semigroups Of Binary Relations Research Articles

Generated sets of the complete semigroup binary relations defined by semilattices of the finite chains

In this article, we study generated sets of the complete semigroups of binary relations defined by X-semilattices unions of the finite chains. We found uniquely irreducible generating set for the given semigroups.

Generated Sets of the Complete Semigroup Binary Relations Defined by Semilattices of the Class Σ<sub>8</sub> (<em>X</em>, <em>n</em> + <em>k</em> + 1)

In this article, we study generated sets of the complete semigroups of binary relations defined by X-semilattices unions of the class Σ8 (X, n + k +1) , and find uniquely irreducible generating set for the given semigroups.

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English

Generating Set of the Complete Semigroups of Binary Relations

Difficulties encountered in studying generators of semigroup of binary relations defined by a complete X -semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup , which are defined by semilattice D, can be generated by the set .

Regular elements of the complete semigroups of binary relations of the class Σ6(X,8)

Abstract In this paper, let X be a finite set, D be a complete X-semilattice of unions and Q = { T 1 , T 2 , T 3 , T 4 , T 5 , T 6 , T 7 , T 8 } $Q=\lbrace T_1,T_2,T_3,T_4,T_5, T_6,T_7,T_8\rbrace $ be an X-subsemilattice of D where T 1 ⊂ T 3 ⊂ T 5 ⊂ T 6 ⊂ T 8 $T_1\subset T_3\subset T_5\subset T_6\subset T_8$ , T 1 ⊂ T 3 ⊂ T 5 ⊂ T 7 ⊂ T 8 $T_1\subset T_3\subset T_5\subset T_7\subset T_8$ , T 2 ⊂ T 3 ⊂ T 5 ⊂ T 6 ⊂ T 8 $T_2\subset T_3\subset T_5\subset T_6\subset T_8$ , T 2 ⊂ T 3 ⊂ T 5 ⊂ T 7 ⊂ T 8 $T_2\subset T_3\subset T_5\subset T_7\subset T_8$ , T 2 ⊂ T 4 ⊂ T 5 ⊂ T 6 ⊂ T 8 $T_2\subset T_4\subset T_5\subset T_6\subset T_8$ , T 2 ⊂ T 4 ⊂ T 5 ⊂ T 7 ⊂ T 8 $T_2\subset T_4\subset T_5\subset T_7\subset T_8$ , T 2 ∖ T 1 ≠ ∅ $T_2\setminus T_1\ne \emptyset $ , T 1 ∖ T 2 ≠ ∅ $T_1\setminus T_2\ne \emptyset $ , T 4 ∖ T 3 ≠ ∅ $T_4\setminus T_3\ne \emptyset $ , T 3 ∖ T 4 ≠ ∅ $T_3\setminus T_4\ne \emptyset $ , T 6 ∖ T 7 ≠ ∅ $T_6\setminus T_7\ne \emptyset $ , T 7 ∖ T 6 ≠ ∅ $T_7\setminus T_6\ne \emptyset $ , T 2 ∪ T 1 = T 3 $T_2\cup T_1=T_3$ , T 4 ∪ T 3 = T 5 $T_4\cup T_3=T_5$ , T 6 ∪ T 7 = T 8 $T_6\cup T_7=T_8$ . Using the characteristic family of sets, the characteristic mapping and base sources of Q, we characterize the class whose elements are each isomorphic to Q. We generate some advanced formulas in order to calculate the number of regular elements α of B X (D) satisfying V ( D , α ) = Q $V(D,\alpha )=Q$ , in an efficient way.

Right Units in Complete Semigroups of Binary Relations

In this paper, we give a full description of right units of the semigroup B X (D), which are defined by semilattices of the class net. For the case where X is a finite set, we derive formulas by calculating the numbers of right units of the corresponding semigroup.

Property of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions

We study properties of right units of complete semigroups of binary relations defined by finite XI-semilattices of unions.

Complete semigroups of binary relations defined by finite XI-semilattices of unions

We give conditions under which the X-semilattice of unions is an XI-semilattice, i.e., let D be a finite X-semilattice of unions. The complete semigroup of binary relation BX(D) is defined by the XI-semilattice of unions if and only if V (D, β) = D for some β ∈ BX(D).

Complete semigroups of binary relations defined by X-semilattices of unions

This paper is devoted to some results from the book “Complete Semigroups of Binary Relations,” which is not published yet. Conditions of divisibility of binary relations, idempotents, regular elements, one-side units, and one-side zeros are described. Finite X-semilattices and unions are characterized. Methods of finding formulas for counting the number of idempotents and regular elements are given.

Regular elements of semigroups B X (D) defined by generalized elementary B X (D)-semilattices of unions

In the paper, the class of complete semigroups of binary relations is considered, each of whose elements is defined by a complete BX(D)-semilattice of unions which belongs to the class of generalized elementary X-semilattices. Regular elements are described for each semigroup of this class.

Idempotents and regular elements of complete semigroups of binary relations

As is known (see [5, 6]), idempotent elements of semigroups play an important role in the investigation of semigroups themselves. Moreover, it is known that any semigroup can be isomorphically embedded into some semigroup of binary relations on some nonempty set X. This has generated great interest in the investigation of idempotent binary relations. Different authors used different approaches to study the construction of these elements and obtained different answers to the question under consideration (see, e.g., [1–3, 7, 8]). In this work, first, we characterize the complete X-semilattices of unions D for which there exist idempotent binary relations such that the set of all their cuts to the elements of D coincides with the given semilattice. Then we give a description of the structure of these idempotent binary relations in the language of limiting elements of some subset of the semilattice D (see Lemma 1 and Theorem 5) and show how we can find all idempotents of the complete group of binary relations BX (D) defined by the complete X-semilattice of unions D (see Theorem 6 and Corollary 4). Moreover, with the help of chains of the semilattice D, we give a rule for constructing the semigroups of idempotent elements of the semigroup BX(D) (see Theorem 7). In Theorem 9, we give a description of the structure of all regular elements of the semigroup BX (D).

On finite X-semilattices of unions

Using a characteristic family of sets, a characteristic mapping, and basis sources of an X-semilattice of unions D, we characterize the class Σ(X, m) consisting of all finite X-semilattices of unions that are isomorphic to a semilattice D given in advance. For a finite set X, the number of elements in the considered class is found. Commutative semigroups of idempotents are known to play a significant role in semigroup theory (see [25, 26]). Moreover, any commutative idempotent semigroup is isomorphic to some X-semilattice of unions (see [26]), whereas X-semilattices play an especially important role in studying many abstract properties of complete semigroups of binary relations (see [1–4, 7–24]).

Maximal Subgroups of Some Classes of Semigroups of Binary Relations

Four classes of complete semigroups of binary relations are considered. It is proved for the semigroups of these classes that maximal subgroups of these semigroups are groups whose order is not greater than two.

One Class of Complete Semigroups of Binary Relations

One Class of Complete Semigroups of Binary Relations

Complete Semigroups of Binary Relations

Complete Semigroups of Binary Relations

Complete Semigroups of Binary Relations Determined by Three-Element X-Chains

Complete Semigroups of Binary Relations Determined by Three-Element X-Chains

Complete Semigroups of Binary Relations Defined by Elementary and Nodal X -Semilattices of Unions

Complete Semigroups of Binary Relations Defined by Elementary and Nodal X -Semilattices of Unions