Categories and General Algebraic Structures with Applications
Volume:1 Issue: 1, Dec 2013

For any archimedeanf-ring A with unit in whichbreakawedgeýý(1−a)leq0 for all ainAý, ýthe following are shown to beý ýequivalentý: ý ý1ý. ýA is isomorphic to the l-ring mathfrakZL of allý ýinteger-valued continuous functions on some frame Lý. 2ý. ýA is a homomorphic image of the l-ring CBbbZ(X)ý ýof all integer-valued continuous functionsý, ýin the usual senseý, ýon some topological space Xý. 3ý. ýFor any family (an)ninomega in A there exists aný ýl-ring homomorphism breakvarphiý:ýCBbbZ(BbbýýZomega)rightarrowA such that varphi(pn)=an for theý ýproduct projections breakpn:BbbZomegarightarrowBbbZ ý. ýThis provides an integer-valued counterpart to a familiar resultý ýconcerning real-valued continuous functionsý

Let L be a completely regular frame and mathcalRL be the ring of continuous real-valued functions on L. We study the frame mathfrakO(Min(mathcalRL)) of minimal prime ideals of mathcalRL in relation to betaL. For IinbetaL, denote by textittextbfOI the ideal alphainmathcalRLmidcozalphainI of mathcalRL. We show that sending I to the set of minimal prime ideals not containing textittextbfOI produces a ∗-dense one-one frame homomorphism betaLtomathfrakO(Min(mathcalRL)) which is an isomorphism if and only if L is basically disconnected.

Recall that a continuous function fcolonXtoY between Tychonoff spaces is proper if and only if the Stone extension fbetacolonbetaXtobetaY takes remainder to remainder, in the sense that fbeta[betaX−X]subseteqbetaY−Y. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism hcolonLtoM to be beta-proper, lambda-proper or upsilon proper in case the lifted homomorphism hbetacolonbetaLtobetaM, hlambdacolonlambdaLtolambdaM or hupsiloncolonupsilonLtoupsilonM takes remainder to remainder. These turn out to be weaker forms of properness. Indeed, every proper homomorphism is beta-proper, every beta-proper homomorphism is lambda-proper, and lambda-properness is equivalent to upsilon-properness. A characterization of beta -proper maps in terms of pointfree rings of continuous functions is that they are precisely those whose induced ring homomorphisms contract free maximal ideals to free prime ideals.

The aim of this paper is to study semigroups possessing E-regular elements, where an element a of a semigroup S is {em E-regular} if a has an inverse acirc such that aacirc,acirca lie in EsubseteqE(S). Where S possesses `enough' (in a precisely defined way) E-regular elements, analogues of Green's lemmas and even of Green's theorem hold, where Green's relations mbox$mathcalR$,el,eh and dee are replaced by artE,eltE,ehtE and widetildemathcalDE. Note that S itself need not be regular. We also obtain results concerning the extension of (one-sided) congruences, which we apply to (one-sided) congruences on maximal subgroups of regular semigroups. If S has an inverse subsemigroup U of E-regular elements, such that EsubseteqU and U intersects every ehtE-class exactly once, then we say that U is an {em inverse skeleton} of S. We give some natural examples of semigroups possessing inverse skeletons and examine a situation where we can build an inverse skeleton in a widetildemathcalDE-simple monoid. Using these techniques, we show that a reasonably wide class of widetildemathcalDE -simple monoids can be decomposed as Zappa-Sz'{e}p products. Our approach can be immediately applied to obtain corresponding results for bisimple inverse monoids.

Semi-regular locales are extensions of the classical semiregular spaces. We investigate the conditions such that semi-regularization is a functor. We also investigate the conditions such that semi-regularization is a reflection or coreflection.

This work is devoted to give a charcaterization of a pomonoid S such that all cyclic S-posets are regular injective.