Usual Partial Order Research Articles

Maximal right congruences on semigroups

A right congruence p on a semigroup S is a relation on S, but also has an algebraic structure. This is perhaps more evident when one views S/p as an Soperand. In this paper we define the notion of isomorphism of two right congruences, and begin a description and classification of the isomorphism classes for those right congruences which are maximal with respect to the usual partial ordering. In particular we characterize completely the twenty-three nonisomorphic right congruences of order two by means of homomorphisms into a particular semigroup. If p is any right congruence on a semigroup S, we define the index of p, written JpJ, to be JS/pJ, the cardinality of S/p. p is said to be right reductive if (ac)p(bc) for all c in S implies apb. For other definitions, the reader is referred to [I]. i. Maximal riqht conqruences. The following result is known and will be useful in the sequel. It

The theory of countable analytical sets

The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n n a largest countable Π n 1 \Pi _n^1 set of reals, C n {\mathcal {C}_n} (this is also true for n n even, replacing Π n 1 \Pi _n^1 by Σ n 1 \Sigma _n^1 and has been established earlier by Solovay for n = 2 n = 2 and by Moschovakis and the author for all even n > 2 n > 2 ). The internal structure of the sets C n {\mathcal {C}_n} is then investigated in detail, the point of departure being the fact that each C n {\mathcal {C}_n} is a set of Δ n 1 \Delta _n^1 -degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, ω \omega -models of analysis, higher-level analogs of the constructible universe, inductive definability, etc.

Metrizable mapping compactifications

Suppose f: X→ f( X)= Y is a continuous function from one completely regular Hausdorff space onto another. There is associated with each possible compactification X̃ of the domain space X a compactification of the mapping f in a unique way; the mapping compactification is called the compactification determined by X̃. The major result of this paper is that if X̃ is metrizable, then the domain of the mapping compactification determined by it is also metrizable if and only if the range Y is. It is also proved that if X 1 and X 2 are two compactifications of X such that X 2⩾ X 1, where ⩾ is the usual partial order on the collection of all compactifications of X, then the compactifications of f determined by X 1 and X 2 are related the same way with respect to the usual partial order on the collection of all compactifications of f.