Categories and General Algebraic Structures with Applications
Volume:7 Issue: 1, Jul 2017

This interview, a co-operative effort of Bernhard Banaschewski and Christopher Gilmour, took place over a few days in December, 2016. It was finalised over coffee and a shared slice of excellent cheesecake at The Botanical Tea Garden, a small, home situated, tea garden in Little Mowbray, Cape Town.

The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical `tangle modality' connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.

Equality algebras were introduced by S. Jenei as a possible algebraic semantic for fuzzy type theory. In this paper, we introduce some types of filters such as (positive) implicative, fantastic, Boolean, and prime filters in equality algebras and we prove some results which determine the relation between these filters. We prove that the quotient equality algebra induced by an implicative filter is a Boolean algebra, by a fantastic filter is a commutative equality algebra, and by a prime filter is a chain, under suitable conditions. Finally, we show that positive implicative, implicative, and Boolean filters are equivalent on bounded commutative equality algebras.

Locally compact Hausdorff spaces and their one-point compactifications are much used in topology and analysis; in lattice and domain theory, the notion of continuity captures the idea of local compactness. Our work is located in the setting of pointfree topology, where lattice-theoretic methods can be used to obtain topological results.
Specifically, we examine here the concept of continuity for partial frames, and compactifications of regular continuous such.
Partial frames are meet-semilattices in which not all subsets need have joins.
A distinguishing feature of their study is that a small collection of axioms of an elementary nature allows one to do much that is traditional for frames or locales. The axioms are sufficiently general to include as examples σ-frames, κ-frames and frames.
In this paper, we present the notion of a continuous partial frame by means of a suitable ``way-below'' relation; in the regular case this relation can be characterized using separating elements, thus avoiding any use of pseudocomplements (which need not exist in a partial frame). Our first main result is an explicit construction of a one-point compactification for a regular continuous partial frame using generators and relations. We use strong inclusions to link continuity and one-point compactifications to least compactifications. As an application, we show that a one-point compactification of a zero-dimensional continuous partial frame is again zero-dimensional.
We next consider arbitrary compactifications of regular continuous partial frames. In full frames, the natural tools to use are right and left adjoints of frame maps; in partial frames these are, in general, not available. This necessitates significantly different techniques to obtain largest and smallest elements of fibres (which we call balloons); these elements are then used to investigate the structure of the compactifications. We note that strongly regular ideals play an important r\^{o}le here. The paper concludes with a proof of the uniqueness of the one-point compactification.

In this paper, we consider the forgetful functor from the category LDcpo of local dcpos (respectively, Dcpo of dcpos) to the category Pos of posets (respectively, LDcpo of local dcpos), and study the existence of its left and right adjoints. Moreover, we give the concrete forms of free and cofree S-ldcpos over a local dcpo, where S is a local dcpo monoid. The main results are: (1) The forgetful functor U : LDcpo −! Pos has a left adjoint, but does not have a right adjoint; (2) The inclusion functor I : Dcpo −! LDcpo has a left adjoint, but does not have a right adjoint; (3) The forgetful functor U : LDcpo-S −! LDcpo has both left and right adjoints; (4) If (S, ·, 1) is a good ldcpo-monoid, then the forgetful functor U: LDcpo-S −! Pos-S has a left adjoint.

In this article we investigate filters of cozero sets for real-valued continuous functions, called coz-filters. Much is known for z-ultrafilters and their correspondence with maximal ideals of C(X). Similarly, a correspondence will be established between coz-ultrafilters and minimal prime ideals ofC(X). We will further notice various properties of coz-ultrafilters in relation to P-spaces and F-spaces. In the last two sections, the collection of cozultrafilters will be topologized, and then compared to the hull-kernel and the inverse topologies placed on the collection of minimal prime ideals of C(X) and general lattice-ordered groups.

Let G = (V,E) be a graph. A subset S of V is a dominating set of G if every vertex in V \ S is adjacent to a vertex in S. A dominating set S is called a secure dominating set if for each v 2 V \ S there exists u 2 S such that v is adjacent to u and S1 = (S \{u})[{v} is a dominating set. If further the vertex u 2 S is unique, then S is called a perfect secure dominating set. The minimum cardinality of a perfect secure dominating set of G is called the perfect secure domination number of G and is denoted by ps(G). In this paper we initiate a study of this parameter and present several basic results.

In this paper, for each lattice-valued map A ! L with some properties, a ring representation A ! RL is constructed. This representation is denoted by c which is an f-ring homomorphism and a Q-linear map, where its index c, mentions to a lattice-valued map. We use the notation a pq = (a − p) ^ (q − a), where p, q 2 Q and a 2 A, that is nominated as interval projection. To get a well-defined f-ring homomorphism c, we need such concepts as bounded, continuous, and Q-compatible for c, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued map cϕ : A ! L for each f-ring homomorphism ϕ : A ! RL. It is proved that cc = cr and cϕ = ϕ, which they make a kind of correspondence relation between ring representations A ! RL and the lattice-valued maps A ! L, where the mapping cr : A ! L is called a realization of c. It is shown that cr = c and crr = cr. Finally, we describe how c can be a fundamental tool to extend pointfree version of Gelfand duality constructed by B. Banaschewski.

The much-studied projectable hull of an ℓ-group G  pG is an essential extension, so that, in the case that G is archimedean with weak unit, “G 2 W”, we have for the Yosida representation spaces a “covering map” Y G Y pG. We have earlier [8] shown that (1) this cover has a characteristic minimality property, and that (2) knowing Y pG, one can write down pG. We now show directly that for A , the boolean algebra in the power set of the minimal prime spectrum Min(G), generated by the sets U(g) = {P 2 Min(G) : g /2 P} (g 2 G), the Stone space SA is a cover of Y G with the minimal property of (1); this extends the result from [1] for the strong unit case. Then, applying (2) gives the pre-existing description of pG, which includes the strong unit description of [1]. The present methods are largely topological, involving details of covering maps and Stone duality.