We show that the free weakly E-ample monoid on a set X is a full submonoid of the free inverse monoid FIM(X) on X . Consequently, it is ample, and so coincides with both the free weakly ample and the free ample monoid FAM(X) on X . We introduce the notion of a semidirect product Y ∗ T of a monoid T acting doubly on a semilattice Y with identity. We argue that the free monoid X∗ acts doubly on t...

Fuzzy logic generalises classical logic; in addition to the latter’s truth values “false” and “true”, the former allows also intermediary truth degrees. The conjunction is, accordingly, interpreted by an operation acting on a chain, making the set of truth degrees into a totally ordered monoid. We present in this chapter two different ways of investigating this type of algebras. We restrict to ...

Let R be an associative unital algebra over a field k, let p be an element of R, and let R′ = R q | pqp = p . We obtain normal forms for elements of R′, and for elements of R′-modules arising by extension of scalars from R-modules. The details depend on where in the chain pR ∩ Rp ⊆ pR ∪ Rp ⊆ pR + Rp ⊆ R the unit 1 of R first appears. This investigation is motivated by a hoped-for application to...

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

The problem of characterization of the structure of MTL-algebras, which form an equivalent algebraic semantics for Monoidal T-norm Based Logic (see [9]) in the sense of Blok and Pigozzi (see [3]), is still far from being solved. Since MTL-algebras are in fact subdirect products of chains, it suffices to investigate only the structure of MTL-chains if we want to characterize the structure of MTL...