Presentations for (singular) partition monoids: a new approach
We give new, short proofs of the presentations for the partition monoid and its singular ideal originally given in the author's 2011 papers in J Alg and IJAC.
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arXiv subjects
James East
Publications and source records attributed to James East.
Presentations for rook partition monoids and algebras and their singular ideals
We obtain several presentations by generators and relations for the rook partition monoids and algebras, as well as their singular ideals. Among other results, we also calculate the minimal sizes of generating sets (some of our presentations use such minimal-size generating sets), and show that the singular part of the rook partition monoid is generated by its idempotents.
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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.
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On groups generated by involutions of a semigroup
An involution on a semigroup S (or any algebra with an underlying associative binary operation) is a function f:S->S that satisfies f(xy)=f(y)f(x) and f(f(x))=x for all x,y in S. The set I(S) of all such involutions on S generates a subgroup C(S)= of the symmetric group Sym(S) on the set S. We investigate the groups C(S) for certain classes of semigroups S, and also consider the question of which groups are isomorphic to C(S) for a suitable semigroup S.
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Idempotent generation in the endomorphism monoid of a uniform partition
Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal P)$ denote the set of all transformations of the finite set $X$ preserving a uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where $m,n\geq2$. We enumerate the idempotents of $\mathcal T(X,\mathcal P)$, and describe the subsemigroup $S=\langle E\rangle$ generated by the idempotents $E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1$ is a direct product of $m$ copies of $\mathcal E_n$, and $S_2$ is a wreath product of $\mathcal T_n$ with $\mathcal T_m\setminus \mathcal S_m$. We calculate the rank and idempotent rank of $S$, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of $\mathcal E_n$.
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Idempotent rank in the endomorphism monoid of a non-uniform partition
We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
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Finite Diagram Semigroups: Extending the Computational Horizon
Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our computational knowledge of them. The generated data set is replete with surprising observations raising many open questions for further theoretical research.
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Enumeration of idempotents in diagram semigroups and algebras
We give a characterisation of the idempotents of the partition monoid, and use this to enumerate the idempotents in the finite partition, Brauer and partial Brauer monoids, giving several formulae and recursions for the number of idempotents in each monoid as well as various $\mathscr R$-, $\mathscr L$- and $\mathscr D$-classes. We also apply our results to determine the number of idempotent basis elements in the finite dimensional partition, Brauer and partial Brauer algebras.
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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 $α,β\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal S_X\cup\{α,β\}$. 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 $\mathcal E_X\cup\{α,β\}$ if and only if it is generated by $\mathcal E_X\cup\mathcal S_X\cup\{α,β\}$. We also classify the pairs $α,β\in\mathcal P_X$ for which $\mathcal P_X$ is generated by $\mathcal E_X\cup\{α,β\}$. 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.
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A Presentation for the Dual Symmetric Inverse Monoid
The dual symmetric inverse monoid $\mathscr{I}_n^*$ is the inverse monoid of all isomorphisms between quotients of an $n$-set. We give a monoid presentation of $\mathscr{I}_n^*$ and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.
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