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abstract: 'In this paper, we determine the relative rank of the semigroup $\T(X,Y)$ of all transformations on a finite chain $X$ with restricted range $Y\subseteq X$ modulo the set $\OP(X,Y)$ of all orientation-preserving transformation in $\T(X,Y)$. Moreover, we state the relative rank of the semigroup $\OP(X,Y)$ modulo the set $\O(X,Y)$ of all order-preserving transformations in $\OP(X,Y)$. In both cases we characterize the minimal relative generating sets.'
author:
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    Ilinka Dimitrova\
    *Faculty of Mathematics and Natural Science*\
    *South-West University “Neofit Rilski”*\
    *2700 Blagoevgrad, Bulgaria*\
    *e-mail: ilinka\_dimitrova@swu.bg*\
      \
    Jörg Koppitz\
    *Institute of Mathematics and Informatics*\
    *Bulgarian Academy of Sciences*\
    *1113 Sofia, Bulgaria*\
    *e-mail: koppitz@math.bas.bg*
title: On relative ranks of finite transformation semigroups with restricted range
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*Key words:* transformation semigroups with restricted range, order-preserving transformations, orientation-preserving transformations, relative rank, relative generating sets.

2010 Mathematics Subject Classification: 20M20\

Introduction and Preliminaries
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Let $S$ be a semigroup. The *rank* of $S$ (denoted by $\rank S$) is defined to be the minimal number of elements of a generating set of $S$. The ranks of various well known semigroups have been calculated [@GH; @GH1; @Howie; @HMcF]. For a set $A \subseteq S$, the *relative rank* of $S$ modulo $A$, denoted by $\rank (S 