Conjugacy in Semigroups Jo˜aoAra´ ujo Universidade Aberta, R. Escola Polit´ ecnica, 147 1269-001 Lisboa, Portugal & Centro de ´ Algebra, Universidade de Lisboa 1649-003 Lisboa, Portugal, mjoao@lmc.fc.ul.pt Michael Kinyon Department of Mathematics, University of Denver Denver, Colorado 80208, mkinyon@math.du.edu Janusz Konieczny Department of Mathematics, University of Mary Washington Fredericksburg, VA 22401, jkoniecz@umw.edu Abstract The action of any group on itself by conjugation and the corresponding conjugacy rela- tion play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper we present a new definition of conju- gacy, which can be applied to an arbitrary semigroup, including semigroups with zero. We characterize this conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups in the case when the set is infinite. 2010 Mathematics Subject Classification . 20M20. Keywords : Conjugation; transformation semigroups; directed graphs; partial homomor- phisms of digraphs 1 Introduction Let G be a group. For elements a,b ∈ G, we say that a is conjugate to b if there exists g ∈ G such that b = g −1 ag. It is clear that this relation is an equivalence on G and that a is conjugate to b if and only if there exists g ∈ G such that ag = gb. Using the latter formulation, one may try to extend the notion of conjugacy to semigroups in the following way: for all elements a and b in a semigroup S, a ∼ 1 b ⇔∃ g∈S 1 ag = gb, (1.1) where S 1 is S with an identity adjoined. (We will write “∼” with various subscripts for possible definitions of conjugacy in semigroups.) In a general semigroup S, the relation ∼ 1 is reflexive and transitive, but not symmetric. However, it is symmetric in any free semigroup. Lallement [11] defines the conjugate elements of a free semigroup S as in (1.1) and shows that the relation ∼ 1 is equal to the following equivalence on the free semigroup S: a ∼ 2 b ⇔∃ u,v∈S 1 a = uv and b = vu. (1.2) In a general semigroup S, the relation ∼ 2 is reflexive and symmetric, but not transitive. 1