Narrow Results

This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramsey's theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids.

For a monoid $M$ and a class of partial actions of $M$, determined by a set, $R$, of identities, we define ${{ℱ}{ℛ}}_R(M)$ to be the universal $M$-generated two-sided restriction monoid with respect to partial actions of $M$ determined by $R$. This is an $F$-restriction monoid which (for a certain $R$) generalizes the Birget–Rhodes prefix expansion ${\tilde{G}}^{{ℛ}}$ of a group $G$. Our main result provides a coordinatization of ${{ℱ}{ℛ}}_R(M)$ via a partial action product of the idempotent semilattice $E({{ℱ}{ℐ}}_R(M))$ of a similarly defined inverse monoid, partially acted upon by $M$. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result.

We show that some properties of ${{ℱ}{ℛ}}_R(M)$ agree well with suitable properties of $M$, such as being cancellative or embeddable into a group. We observe that if $M$ is an inverse monoid, then ${{ℱ}{ℐ}}_{R_s}(M)$, the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion $M^{\mathrm{\text{pr}}}$. This gives a presentation of $M^{\mathrm{\text{pr}}}$ and leads to a model for ${{ℱ}{ℛ}}_{R_s}(M)$ in terms of the known model for $M^{\mathrm{\text{pr}}}$.

An immersion $f:{𝒟}\to {𝒞}$ between $\mathrm{\Delta }$-complexes is a $\mathrm{\Delta }$-map that induces injections from star sets of ${𝒟}$ to star sets of ${𝒞}$. We study immersions between finite-dimensional connected $\mathrm{\Delta }$-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersions into $2$-dimensional $CW$-complexes.

The paper deals with certain breaking processes using a compatible partition of a monoid. We introduce the broken Dirichlet convolution and the broken bicyclic semigroup. Both have a common origin and are introduced by the same elementary categorical construction.