Categories and General Algebraic Structures with Applications
Volume:9 Issue: 1, Jul 2018

The category of the title is called W. This has all free objects F(I) (I a set). For an object class A, HA consists of all homomorphic images of A-objects. This note continues the study of the H-closed monoreflections (R,r) (meaning HR=R), about which we show ({\em inter alia}): A∈A if and  only if A is a countably up-directed union from H{rF(ω)}. The meaning of this is then analyzed for two important cases: the maximum essential monoreflection r=c3, where c3F(ω)=C(\RRω), and C∈H{c(\RRω)} means C=C(T), for T a closed subspace of \RRω; the epicomplete, and maximum, monoreflection, r=β, where βF(ω)=B(\RRω), the Baire functions, and E∈H{B(\RRω)} means E is {\em an} epicompletion (not ``the'') of such a C(T).

Let £ be a 0-distributive lattice with the least element 0, the greatest element 1, and Z(£) its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by T(G(£)). It is the graph with all elements of £ as vertices, and for distinct x,y∈£, the vertices x and y are adjacent if and only if x∨y∈Z(£). The basic properties of the graph T(G(£)) and its subgraphs are studied. We investigate the properties of the total graph of 0-distributive lattices as diameter, girth, clique number, radius, and the  independence number.

Given a pseudomonad T on a 2-category B, if a right biadjoint A→B has a lifting to the pseudoalgebras A→Ps-T-Alg then this lifting is also right biadjoint provided that A has codescent objects. In this paper, we give  general results on lifting of biadjoints. As a consequence, we get a biadjoint triangle theorem which, in particular, allows us to study triangles involving the 2-category of lax algebras, proving analogues of the result described above. In the context of lax algebras, denoting by ℓ:Lax-T-Alg→Lax-T-Algℓ the inclusion, if R:A→B is right biadjoint and has a lifting J:A→Lax-T-Alg, then ℓ∘J is right biadjoint as well provided that A has some needed weighted bicolimits. In order to prove such result, we study descent objects and lax descent objects. At the last section, we study direct consequences of our theorems in the context of the 2-monadic approach to coherence.

Let RL be the ring of real-valued continuous functions on a frame L as the pointfree  version of C(X), the ring of all real-valued continuous functions on a topological space X. Since Cc(X) is the largest subring of C(X) whose elements have countable image, this motivates us to present the pointfree  version of Cc(X).
The main aim of this paper is to present the pointfree version of image of real-valued continuous functions in RL. In particular, we will introduce the pointfree version of the ring Cc(X). We define a relation from RL into the power set of R, namely overlap. Fundamental properties of this relation are studied. The relation overlap is a pointfree version of the relation defined as Im(f)⊆S for every continuous function f:X→R and S⊆R.

Generalising Nachbin's theory of ``topology and order'', in this paper we   continue the study of quantale-enriched categories equipped with a compact   Hausdorff topology. We compare these \V-categorical compact Hausdorff spaces   with ultrafilter-quantale-enriched categories, and show that the presence of a   compact Hausdorff topology guarantees Cauchy completeness and (suitably   defined) codirected completeness of the underlying quantale enriched category.

In this paper, we have focused to study convex L-subgroups of an L-ordered group. First, we introduce the concept of a convex L-subgroup and a convex L-lattice subgroup of an L-ordered group and give some examples. Then we find some properties and use them to construct convex L-subgroup generated by a subset S of an L-ordered group G . Also, we generalize a well known result about the set of all convex subgroups of a lattice ordered group and prove that C(G), the set of all convex L-lattice subgroups of an L-ordered group G, is an L-complete lattice on height one. Then we use these objects to construct the quotient L-ordered groups and state some related results.