Dense Relations Are Determined by Their Endomorphism Monoids

Abstract

We introduce the class of dense relations on a set X and prove that for any finitary or infinitary dense relation Ω on X, the relational system (X, Ω) is determined up to semi-isomorphism by the monoid End (X, Ω) of endomorphisms of (X, Ω). In the case of binary relations, a semi-isomorphism is an isomorphism or an anti-isomorphism.2000 Mathematics Subject Classification: 20M20, 20M15.

Automorphisms of endomorphism semigroups of reflexive digraphs

“…We proved that U S (Z n ) = U ± S (Z n ), and the result follows by Theorem 4.12. {(1, 2), (1,8), (1,11), (2,4), (2,7), (2,8), (4,8), (4,14), (7,11), . .…”

Полугруппы Эндотопизмов Отношения Эквивалентности

“…Например, Л. Б. Шнеперман (см. [6]) показал, что результат Глускина невозможно перенести на класс всех рефлексивных бинарных отношений, в то время как в [7] упомянутый результат был распространен на так называемые плотные отношения, а в [8] -на некоторый подкласс рефлексивных бинарных отношений. Подобные результаты для определенных µ-арных отношений были получены Б. В. Поповым (см.…”

Automorphism Groups of Circulant Digraphs With Applications to Semigroup Theory

“…The description of automorphisms has a long tradition in mathematics. Regarding automorphisms of semigroups, the pioneering work of Schreier [36] and Mal'cev [32] -proving that the group of automorphisms of the full transformation monoid T n is isomorphic to the symmetric group S n -was followed by a long sequence of similar descriptions (see [1,4,5,6,8,9,7,10,11,12,22,25,26,27,30,31,33,37,38,39,45] and the references therein). The effort to find the automorphisms of transformation semigroups containing all constants culminated in 1972 with the description, provided by Vazenin [41], of the automorphisms of the endomorphism monoid End(Γ), where Γ is a reflexive digraph containing an edge that is not contained in a cycle (of length at least 2).…”