On the ideal structure of certain semirings and compactification of topological spaces

Horne, J. G.

doi:10.1090/s0002-9947-1959-0108714-9

Cited by 8 publications

(2 citation statements)

References 22 publications

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“…Such structure spaces are compact and T 1 . All algebraic structures have different families of ideals, so these families represent the topologies intrinsically [4]. A natural way in which to study this situation is the semigroups.…”

Section: Charterizing M-regular and M-intraregular Semigroupsmentioning

confidence: 99%

“…A natural way in which to study this situation is the semigroups. Detailed procedures of defining the topologies in semigroups and other algebraic structures are given in [8] and [4]. In the beginning of this section, we define the comaximal m-bi ideals in semigroups, which will be used later in the section.…”

Section: Charterizing M-regular and M-intraregular Semigroupsmentioning

confidence: 99%

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Exploring the Algebraic and Topological Properties of Semigroups Through Their Prime $m$-Bi Ideals

MUNIR,

KAUSAR,

ANITHA

et al. 2024

KgJMat

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We introduce the concepts of the prime m-bi ideal and their associated types in the semigroups. Diﬀerent characterizations of the semigroups using these m-bi ideals are presented. The forms of the topologies induced by the prime and strongly prime m-bi ideals in the semigroups are also explored. The result shows that either both the conditions of m-regularity and m-intraregularity or existence of pairwise comaximal m-bi ideals in a semigroup is necessary for strongly prime m-bi ideals to induce a topology; whereas the existence of pairwise comaximal m-bi ideals is necessary for the prime m-bi ideals to induce topology on the semigroups. We concluded that the prime m-bi ideals are as important to study the semigroups as the prime bi ideals.

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Section: Charterizing M-regular and M-intraregular Semigroupsmentioning

confidence: 99%

Section: Charterizing M-regular and M-intraregular Semigroupsmentioning

confidence: 99%

Exploring the Algebraic and Topological Properties of Semigroups Through Their Prime $m$-Bi Ideals

MUNIR,

KAUSAR,

ANITHA

et al. 2024

KgJMat

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Boolean ?-algebras and quasiopen mappings

Fedorchuk¹

1974

Sib Math J

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u-Isomorphic semigroups of continuous functions

Császár

1986

Acta Math Hung

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A. CSASZAR (Budapest), member of the Academy 0. Introduction. Let X be a topological space, and denote by C(X) the set of all real-valued continuous functions defined on X, by C*(X) the subset of C(X) composed of bounded functions. Both C(X) and C*(X) can be considered as a ring under pointwise addition and multiplication of functions, or as a semigroup under pointwise multiplication. For a completely regular Hausdorff space X, let fix and vX denote the (~ech Stone compactification and the Hewitt realcompactification of X, respectively (see e. g.[2]).The following propositions are well-known for completely regular Hausdortf spaces X and Y:A. If C(X) and C(Y) are ring isomorphic, then vX and vY are homeomorphic. B. If C*(X) and C* (Y) are ring isomorphic, then fix and fly are homeomorphic.It is also known that, both in Propositions A and B, ring isomorphy can be replaced by semigroup isomorphy. Moreover, for A, an essentially stronger result is valid. In order to formulate it, let us recall (cf.[1]) that, in a semigroup S, we write at>b for a, bCS, iffthere exists an element c~S such that a=cb; for two semigroups $1 and S~ and the respective relations ~>i and t>~, a bijection ~p: SI-*S~ is said to be a d-isomorphism iff a~,-l b ~, q~(a)~>~q~(b).C. (ef.[1], Theorem 3). For two completely regular Hausdorff spaces X and Y, if C(X) and C(Y) are d-isomorphic, then vX and vY are homeomorphic. ~

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On the ideal structure of certain semirings and compactification of topological spaces

Cited by 8 publications

References 22 publications

Exploring the Algebraic and Topological Properties of Semigroups Through Their Prime $m$-Bi Ideals

Exploring the Algebraic and Topological Properties of Semigroups Through Their Prime $m$-Bi Ideals

Boolean ?-algebras and quasiopen mappings

u-Isomorphic semigroups of continuous functions

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