On relative ranks of finite transformation semigroups with restricted range

I. Dimitrova; J. Koppitz

doi:10.37863/umzh.v73i5.288

I. Dimitrova South-West University “Neofit Rilski”, Blagoevgrad, Bulgaria https://orcid.org/0000-0002-6078-5968
J. Koppitz Institute of Mathematics and Informatics Bulgarian Academy of Sciences, Sofia, Bulgaria

DOI: https://doi.org/10.37863/umzh.v73i5.288

Keywords: transformation semigroups with restricted range, orderpreserving transformations, orientation-preserving transformations, relative rank, relative generating sets

Abstract

UDC 512.5

We determine the relative rank of the semigroup ${\scr T }(X,Y)$ of all transformations on a finite chain $X$ with restricted range $Y\subseteq X$ modulo the set ${\scr OP }(X,Y)$ of all orientation-preserving transformations in ${\scr T }(X,Y).$ Moreover, we state the relative rank of the semigroup ${\scr OP }(X,Y)$ modulo the set ${\scr O}(X,Y)$ of all order-preserving transformations in ${\scr OP}(X,Y).$ In both cases we characterize the minimal relative generating sets.

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