For a closure space (P,φ) with φ(∅) = ∅, the closures of open subsets of P , called the regular closed subsets, form an ortholattice Reg(P,φ), extending the poset Clop(P,φ) of all clopen subsets. If (P,φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, ...

The application of ideas from universal algebra to computer science has long been a major theme of Joseph Goguen’s research, perhaps even the major theme. One strand of this work concerns algebraic datatypes. Recently there has been some interest in what one may call algebraic computation types. As we will show, these are also given by equational theories, if one only understands the notion of ...

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...

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 ...

Recently W. Holliday gave a choice-free construction of canonical extension boolean algebra B as the regular open subsets Alexandroff topology on poset proper filters B. We make this point-free by replacing space with free frame $$\mathcal {L}_B$$ generated bounded meet-semilattice all (ordered reverse inclusion) and prove that booleanization is Our main result generalizes approach to category ...

Each factor semigroup of a free restriction (ample) semigroup over a congruence contained in the least cancellative congruence is proved to be embeddable into a W -product of a semilattice by a monoid. Consequently, it is established that each restriction semigroup has a proper (ample) cover embeddable into such a W -product.

We characterize free poset algebras F (P ) over partially ordered sets and show that they can be represented by upper semi-lattice algebras. Hence, the uniqueness, in decomposition into normal form, using symmetric difference, of non-zero elements of F (P ) is established. Moreover, a characterization of upper semilattice algebras that are isomorphic to free poset algebras is given in terms of ...