Related papers: Partial transformation monoids preserving a unifor…
We build on the recent characterisation of congruences on the infinite twisted partition monoids $\mathcal{P}_{n}^\Phi$ and their finite $d$-twisted homomorphic images $\mathcal{P}_{n,d}^\Phi$, and investigate their algebraic and…
Malcev described the congruences of the monoid $T_n$ of all full transformations on a finite set $X_n=\{1, \dots,n\}$. Since then, congruences have been characterized in various other monoids of (partial) transformations on $X_n$, such as…
Let G1, G2, ... be a sequence of almost simple groups and construct a sequence (Wi) of wreath products via W1 = G1 and, for each i > 1, Wi+1 = Gi+1 wr Wi via the regular action of each Gi. We determine the minimum number d(Wi) of generators…
A smallest generating set of a semigroup is a generating set of the smallest cardinality. Similarly, an irredundant generating set $X$ is a generating set such that no proper subset of $X$ is also a generating set. A semigroup $S$ is…
We calculate the number of unary clones (submonoids of the full transformation monoid) containing the permutations, on an infinite base set. It turns out that this number is quite large, on some cardinals as large as the whole clone…
The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If…
A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard…
We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner…
We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering and for practically investigating these monoids. The approach is based on subword reversing, a general method of…
An affine semigroup is a finitely generated subsemigroup of $(\mathbb Z_{\ge 0}^d, +)$, and a numerical semigroup is an affine semigroup with $d = 1$. A growing body of recent work examines shifted families of numerical semigroups, that is,…
A variety is finitely universal if its lattice of subvarieties contains an isomorphic copy of every finite lattice. Examples of finitely universal varieties of semigroups have been available since the early 1970s, but it is unknown if there…
A function on an algebra is congruence preserving if, for any congruence, it maps congruent elements to congruent elements. We show that, on a free monoid generated by at least 3 letters, a function from the free monoid into itself is…
Wilcox has considered a twisted semigroup algebra structure on the partition algebra $\mathbb{C}A_k(n)$, but it appears that there has not previously been any known basis that gives $\mathbb{C}A_k(n)$ the structure of a "non-twisted"…
We study the homology of pointed sets over a partially commutative monoid.
Given a sequence of $(G_i)_{i \in \N}$ of finite transitive groups of degree $n_i$, let $W_\infty$ be the inverse limit of the iterated permutational wreath products of the first m groups. We prove that $W_\infty$ is (topologically)…
We introduce the depth parameters of a finite semigroup, which measure how hard it is to produce an element in the minimum ideal when we consider generating sets satisfying some minimality conditions. We estimate such parameters for some…
In this paper, first we introduce a quantity called a partition function for a quiver mutation sequence. The partition function is a generating function whose weight is a $q$-binomial associated with each mutation. Then, we show that the…
This paper deals with the algebraic structure of the sequence of harmonics when combined with equal temperaments. Fractals and the golden ratio appear surprisingly on the way. The sequence of physical harmonics is an increasingly enumerable…
Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of…
We prove that the set of orders of finite quotients of a finitely generated group has natural density 0, 1/2 or 1, and characterise when each of these cases occurs. We apply this to show that the sets of orders of various families of…