For certain a, b ∈ N, the Arithmetic Congruence Monoid M(a, b) is a multiplicatively closed subset of N given by {x ∈ N : x ≡ a (mod b)}∪{1}. An irreducible in this monoid is any element that cannot be factored into two elements, each greater than 1. Each monoid element (apart from 1) may be factored into irreducibles in at least one way. The elasticity of a monoid element (apart from 1) is the...

The dual symmetric inverse monoid I ∗ n is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of I ∗ n and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.

We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For whose local state space has elements, this approach yields unified treatment the well-known additive cancellative dualities. spaces three or more we discover several new

We introduce and study the partial singular braid monoid PSBn, a monoid that contains both the inverse braid monoid IBn and the singular braid monoid SBn. Our main results include a characterization of Green’s relations, a presentation in terms of generators and relations, and a proof that PSBn embeds in the semigroup algebra C[IBn].

We prove that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of End(R), the endomorphism monoid of the infinite random graph R. Consequently, End(R) embeds each countable monoid, satisfies no nontrivial monoid identity, and has an undecidable universal theory.

We generalize the notion of a coloring complex graph to linearized combinatorial Hopf monoids. determine when monoid has such construction, and discover some inequalities that are satisfied by quasisymmetric function invariants associated monoid. show collection all complexes forms monoid, which is terminal object in category monoids with convex characters. also study several examples

We prove that a trace monoid embeds into the queue monoid if and only if it embeds into the direct product of two free monoids. We also give a decidable characterization of these trace monoids.

A monoid hypersurface is an irreducible hypersurface of degree d which has a singular point of multiplicity d−1. Any monoid hypersurface admits a rational parameterization, hence is of potential interest in computer aided geometric design. We study properties of monoids in general and of monoid surfaces in particular. The main results include a description of the possible real forms of the sing...