The semilattice relevant logics ∪R, ∪T, ∪RW, and ∪TW (slightly different from the orthodox relevant logics R, T, RW, and TW) are defined by “semilattice models” in which conjunction and disjunction are interpreted in a natural way. In this paper, we prove the equivalence between “LK-style” and “LJ-style” labelled sequent calculi for these logics. (LKstyle sequents have plural succedents, while ...

The main purpose of this paper is to show that a regular semigroup S is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups and that this holds if and only if 3) is a congruence on S. It is also shown that a quasiregular semigroup 5 which is a rectangular band of bisimple semigroups is itself bisimple. In [3, Theorem 4.4] it was shown that a semigroup S is a ...

We show that no finite set of first-order axioms can define the class of representable semilattice-ordered monoids.

We exhibit some new connections between structure of an information system and its corresponding semilattice of equivalence relations. In particular, we investigate dependency properties and introduce a partial ordering of information systems over a fixed object set U which reflects the sub–semilattice relation on the set of all equivalence relations on U.

We show that a locally finite variety which omits abelian types is self–rectangulating if and only if it has a compatible semilattice term operation. Such varieties must have type–set {5 }. These varieties are residually small and, when they are finitely generated, they have definable principal congruences. We show that idempotent varieties with a compatible semilattice term operation have the ...

A specialization semilattice is a structure which can be embedded into $(\mathcal P(X), \cup, \sqsubseteq )$, where $X$ topological space, $ x y$ means $x \subseteq Ky$, for $x,y X$, and $K$ closure in $X$. Specialization semilattices posets appear as auxiliary structures many disparate scientific fields, even unrelated to topology. In general, not expressible semilattice. On the other hand, we...

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 ...

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...