Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

On quasi completely regular semirings

Abstract: We extend the concepts of a completely π-regular semigroup and a GV semigroup to semirings and find a semiring analogue of a structure theorem on GV semigroups. We also show that a semiring S is quasi completely regular if and only if S is an idempotent semiring of quasi skew-rings.

Ramsey algebras and the existence of idempotent ultrafilters

Abstract: Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open problem posed by Carlson whether every Ramsey algebra has an idempotent ultrafilter. This paper develops a general framework to study idempotent ultrafilters. Under certain countable setting, the main result roughly says that every nondegenerate Ramsey algebra has a nonprincipal idempotent ultrafilter in some nontrivial countable field of sets. This amounts to a positive result that addresses Carlson's question in some way.

Remarks on primitive idempotents in compact semigroups with zero

Abstract: We recall that a mob is a Hausdorff space together with a continuous associative multiplication. A nonempty subset A of a mob X is a submob if AA CA. This note consists of an amplification of results of Numakura dealing with primitive idempotents in a compact mob X with zero (see definitions below). We discuss the properties of certain "fundamental" sets determined by primitive idempotents, namely the sets XeX, Xe, eX, and eXe, where e is a primitive idempotent. These are, respectively, the smallest (two-sided, left, right, bi-) ideal containing e. Included in Theorem 1 is a characterization of a primitive idempotent in terms of its "fundamental" sets. There then follow some remarks on the structure of the smallest ideal containing the set of all primitive idempotents. Finally, if e is a nonzero primitive idempotent of the compact connected mob X with zero, then the set of nilpotent elements of X is dense in each of the "fundamental" sets determined by e. It is a pleasure to acknowledge the advice and helpful criticism of W. M. Faucett and A. D. Wallace. We shall assume throughout most of this note that X is a compact mob with zero (0). For aCzX we denote by F(a) the closure of the set of positive powers of a, and by K(a) the minimal (closed) ideal of F(a). K(a) is known to be a (topological) group and consists of the cluster points of the set of powers of a (1[3; 5]; these results depend only on the compactness of F(a)). Also F(a) contains exactly one idempotent, e, and if e =0 then the powers of a converge to 0. An element a is termed nilpotent if its powers converge to 0, and we denote by N the set of all nilpotent elements of X. A subset A of X is termed nil if A CN. An idempotent e of X is primitive if g =g2CeXe implies g =0 or g = e. Recall that a subset A of X is a bi-ideal if (1) AACA and (2) AXACA [2; 3].

Localization of hoop - algebras

Abstract: In this paper, we use topology and idempotent subalgebra to define local- ization of hoop-algebras. Then we define the notion of hoop-algebra fractions relative to a ∧-closed system and we prove that the hoop-algebra of fractions relative to an ∧-closed system are hoop- algebra of localization.