Square roots in finite full transformation semigroups

Variants of finite full transformation semigroups

The variant of a semigroup S with respect to an element a ∈ S, denoted S, is the semigroup with underlying set S and operation ? defined by x ? y = xay for x, y ∈ S. In this article, we study variants T a X of the full transformation semigroup TX on a finite set X. We explore the structure of T a X as well as its subsemigroups Reg(T a X) (consisting of all regular elements) and E X (consisting ...

Elements with Square Roots in Finite Groups

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

Faster Square Roots in Annoying Finite Fields

Green's Relations in Finite Transformation Semigroups

We consider the complexity of Green’s relations when the semigroup is given by transformations on a finite set. Green’s relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of mag...

UNIVERSAL COMPACTIFICATIONS OF TRANSFORMATION SEMIGROUPS

We extend the notion of semigroup compactification to the class of transformation semigroups, and determine the compactifications which are universal with respect to some topological properties.