On monoids of endomorphisms of a cycle graph

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities

The endomorphisms monoids of graphs of order n with a minimum degree n − 3

We characterize the endomorphism monoids, End(G), of the generalized graphs G of order n with a minimum degree n − 3. Criteria for regularity, orthodoxy and complete regularity of those monoids based on the structure of G are given

Moment categories and operads

A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment structures on $\Gamma$, $\Delta$ and $\Theta_n$ respectively. There is an analog of Baez-Dolan's plus construction taking a unital moment category $\mathbb{C}$ to a unital hypermoment category $\mathbb{C}^+$. Under this construction, $\mathbb{C}$-operads get identified with $\mathbb{C}^+$-monoids, i.e. presheaves on $\mathbb{C}^+$ satisfying Segal-like conditions strictly. We show that the plus construction of Segal's category $\Gamma$ embeds into the dendroidal category $\Omega$ of Moerdijk-Weiss.Comment: Introduction and Bibliography extended. Definition of reduced dendrix corrected. Proofs of Section 3 amended. Two appendices adde

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