Infinite partition monoids

Let $\mathcal P_X$ and $\mathcal S_X$ be the partition monoid and symmetric group on an infinite set $X$. We show that $\mathcal P_X$ may be generated by $\mathcal S_X$ together with two (but no fewer) additional partitions, and we classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal S_X\cup\{\alpha,\beta\}$. We also show that $\mathcal P_X$ may be generated by the set $\mathcal E_X$ of all idempotent partitions together with two (but no fewer) additional partitions. In fact, $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$ if and only if it is generated by $\mathcal E_X\cup\mathcal S_X\cup\{\alpha,\beta\}$. We also classify the pairs $\alpha,\beta\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal E_X\cup\{\alpha,\beta\}$. Among other results, we show that any countable subset of $\mathcal P_X$ is contained in a $4$-generated subsemigroup of $\mathcal P_X$, and that the length function on $\mathcal P_X$ is bounded with respect to any generating set

Variants of finite full transformation semigroups

The variant of a semigroup S with respect to an element a in S, denoted S^a, is the semigroup with underlying set S and operation * defined by x*y=xay for x,y in S. In this article, we study variants T_X^a of the full transformation semigroup T_X on a finite set X. We explore the structure of T_X^a as well as its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of Reg(T_X^a). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idempotent) generating sets of the minimal possible size.Comment: 25 pages, 6 figures, 1 table - v2 includes a couple more references - v3 changes according to referee comments (to appear in IJAC