Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

Abstract

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup $PTD(B_n)$ of partial contracting transformations of a Boolean, the semigroup $TD(B_n)$ of full contracting transformations of a Boolean, and the inverse semigroup $ISD(B_n)$ of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For $PTD(B_n)$ and $TD(B_n)$, the capacity of a maximal nilpotent subsemigroup is calculated. For $ISD(B_n)$, we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small $n$. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree $k$ and determine the number of elements and subsemigroups for some special cases.