### Semilattice graphs, matrix representations of idempotent semigroups and Waterloo automaton semilattices

#### Abstract

Semigroups are a numerous class of algebras due to the fact that only one axiom of associativity of an operation must hold in a semigroup. So, for example, there are 836021 semigroups of the seventh order, and already 1843120128 semigroups of the eighth order. For comparison, there is only one group of order seven, 5 groups of order eight. It is convenient to define semigroups of small orders by the Cayley table, but even in this case it can be quite difficult to get some idea of the structure of the semigroup. In the case when it is possible to obtain an exact matrix representation of a semigroup, the structure of the semigroup becomes clear, since matrices are a well-studied semigroup with a multiplication operation defined in a certain way. As a rule, it is difficult to find the exact matrix representation of a semigroup; first, it is convenient to find a representation of this semigroup by a transformation semigroup. In this case, in what follows, instead of the elements of the original semigroup, we consider the corresponding transformations. The article considers the semigroups of idempotents S, the corresponding semigroups of transformations, as well as the connection between the ranks of these transformations and the presence of an exact matrix representation of the original semigroups. Since any semigroup of idempotents is a semilattice of rectangular semigroups, we first consider the exact matrix representations of a finite semilattice and an arbitrary rectangular semigroup, then describe the exact representation of the semilattice of right zero semigroups in the case when the right shifts of all elements of such a semigroup have a finite rank. The second part of this paper presents the results of a study of semilattices on a set of grid subsets of the Waterloo automaton.

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