We study the free objects in variety of semigroups and monoids generated by monoid all n×n upper triangular matrices over a commutative semiring. obtain explicit representations these, as multiplicative subsemigroups quiver algebras polynomial semirings. A case particular interest is where n=2 semiring tropical semifield, coincides with that bicyclic (or equivalently, monogenic inverse monoid);...

In contrast to the semilattice of groups case, an inverse semigroup S which is the union of strongly ^-reflexive inverse subsemigroups need not be strongly £-reflexive. If, however, the union is saturated with respect to the Green's relation <3), and in particular if the union is a disjoint one, then 5 is indeed strongly £-reflexive. This is established by showing that fy -saturated inverse sub...

The paper presents general machinery for extending a duality between complete, cocomplete categories to a duality between corresponding categories of semilattice representations (i.e. sheaves over Alexandrov spaces). This enables known dualities to be regularized. Among the applications, regularized Lindenbaun-Tarski duality shows that the weak extension of Boolean logic (i.e. the semantics of ...

Abstract It is known that every non-universal self-full degree in the structure of degrees computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using degrees. We show considerably more can be done by staying entirely inside collection non-self-full pos...

A nearlattice is a meet semilattice A in which every initial segment Ap := {x : x ≤ p} happens to be a join semilattice (hence, a lattice) with respect to the natural ordering of A. If all lattices Ap are distributive, the nearlattice itself is said to be distributive. It is known that a distributive nearlattice can be represented as a nearlattice of sets. We call an algebra (A,∧,∨) of type (2,...

We present a method for constructing factorizable inverse monoids (FIMs) from a group and a semilattice, and show that every FIM arises in this way. We then use this structure to describe a presentation of an arbitrary FIM in terms of presentations of its group of units and semilattice of idempotents, together with some other data. We apply this theory to quickly deduce a well known presentatio...

Abstract. Let L be a lattice and M a bounded distributive lattice. Let ConL denote the congruence lattice of L, P (M) the Priestley dual space of M , and L (M) the lattice of continuous order-preserving maps from P (M) to L with the discrete topology. It is shown that Con(L ) ∼= (ConL) P (ConM) Λ , the lattice of continuous order-preserving maps from P (ConM) to ConL with the Lawson topology. V...

This paper studies some properties of the so-called semilattice-based logics (which are defined in a standard way using only the order relation from a variety of algebras that have a semilattice reduct with maximum) under the assumption that its companion assertional logic (defined from the same variety of algebras using the top element as representing truth) is algebraizable. This describes a ...

Contexts where entity descriptions belong to a meet-semilattice are considered. When the entity set is finite, we show that nonempty extensions of concepts assigned to such contexts coincide, casewise, with strong or weak clusters associated with some pairwise or multiway dissimilarity measure. Moreover, by duality principle, a similar result holds when entity descriptions belong to a join-semi...