While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non...

I define the concepts of multifuncoid (and completary multifuncoid) and upgrading. Then I conjecture that upgrading of certain multifuncoids are multifuncoids (and that upgrading certain completary multifuncoids are completary multifuncoids). I have proved the conjectures for n 6 2. The main conjecture from this article is now proved in the article “Multidimensional Funcoids“ This short article...

Some recent results provide su,cient conditions for complete lattices of closure operators on complete lattices, ordered pointwise, to be pseudocomplemented. This paper gives results of pseudocomplementation in the more general setting of closure operators on mere posets. The following result is 0rst proved: closure operators on a meet-continuous meet-semilattice form a pseudocomplemented compl...

We introduce the notion of definite inequality constraints involving monotone functions in a finite meet-semJlattice, generalizing the logical notion of Horn-clauses, and we give a linear time algorithm for deciding satisfiability. We characterize the expressiveness of the framework of definite constraints and show that the algorithm uniformly solves exactly the set of all meet-closed relationa...

Let A := (A,→, 1) be a Hilbert algebra. The monoid of all unary operations on A generated by operations αp : x → (p → x), which is actually an upper semilattice w.r.t. the pointwise ordering, is called the adjoint semilattice of A. This semilattice is isomorphic to the semilattice of finitely generated filters of A, it is subtractive (i.e., dually implicative), and its ideal lattice is isomorph...

A 〈∨, 0〉-semilattice is ultraboolean, if it is a directed union of finite Boolean 〈∨, 0〉-semilattices. We prove that every distributive 〈∨, 0〉-semilattice is a retract of some ultraboolean 〈∨, 0〉-semilattice. This is established by proving that every finite distributive 〈∨, 0〉-semilattice is a retract of some finite Boolean 〈∨, 0〉-semilattice, and this in a functorial way. This result is, in tu...

This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which we make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting there our axiomatization of partial frames, which we call $sels$-frames, we added structure, in th...

The aim of this paper is to introduce the notion of commutative pseudo BE-algebras and investigate their properties.We generalize some results proved by A. Walendziak for the case of commutative BE-algebras.We prove that the class of commutative pseudo BE-algebras is equivalent to the class of commutative pseudo BCK-algebras. Based on this result, all results holding for commutative pseudo BCK-...

We find a distributive (∨, 0, 1)-semilattice Sω1 of size א1 that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: — There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with ...

We define the tensor product A ® S for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see 14], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice A <3 B is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone sp...