Semigroup of transformations with restricted partial range: Regularity, abundance and some combinatorial results

2023/06/01 by Jiulin Jin, Jin, Jiulin, Taijie You +1
Computer Science · Mathematics · #semigroups and automata theory #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2306.00822

Abstract

Suppose that X be a nonempty set. Denote by T(X) the full transformation semigroup on X. For \varnothing ≠ Z⊆ Y⊆ X, let T(X,Y,Z)=\α∈ T(X): Yα⊆ Z \. Then T(X,Y,Z) is a subsemigroup of T(X). In this paper, we characterize the regular elements of the semigroup T(X,Y,Z), and present a necessary and sufficient condition under which T(X,Y,Z) is regular. Furthermore, we investigate the abundance of the semigroup T(X,Y,Z) for the case Z\subsetneq Y\subsetneq X. In addition, we compute the cardinalities of T(X,Y,Z), \rm Reg(T(X,Y,Z)) and \rm E(T(X,Y,Z)) when X is finite, respectively.

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