Homomorphisms of implicative semigroups

Chan, M. W.; Shum, K. P.

doi:10.1007/bf02573537

Cited by 28 publications

(24 citation statements)

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“…To deeply study implicative semigroups, it is undoubtedly necessary to establish more complete theory of ordered filters for it. This paper continues investigations of implicative semigroups and of their ordered filters which were started in the general case by Chan and Shum [3]. We introduce a special set in an implicative semigroup.…”

mentioning

confidence: 65%

“…For the general development of implicative semilattice theory, the ordered filters play an important role which is shown by Nemitz [6]. To be motivated by this, Chan and Shum [3] established some elementary properties, and constructed quotient structure of implicative semigroups via ordered filters. Jun et al [5] discussed ordered filters of implicative semigroups.…”

Section: Introductionmentioning

confidence: 99%

“…The notions of implicative semigroup and ordered filter were introduced by Chan and Shum [3]. The first is a generalization of implicative semilattice (see Nemitz [6] and Blyth [2]) and has a close relation with implication in mathematical logic and set theoretic difference (see Birkhoff [1] and Curry [4]).…”

Section: Introductionmentioning

confidence: 99%

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Some results on ordered filters of implicative semigroups

Jun

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. We introduce a special set in an implicative semigroup. Using it, an equivalent condition of an ordered filter is stated. We prove that an ordered filter can be represented by the union of such sets. [5] discussed ordered filters of implicative semigroups. To deeply study implicative semigroups, it is undoubtedly necessary to establish more complete theory of ordered filters for it. This paper continues investigations of implicative semigroups and of their ordered filters which were started in the general case by Chan and Shum [3]. We introduce a special set in an implicative semigroup. By using this set we give an equivalent condition of an ordered filter. We prove that an ordered filter can be represented by the union of such sets.

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mentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

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Some results on ordered filters of implicative semigroups

Jun

2001

International Journal of Mathematics and Mathematical Sciences

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Section: Introductionmentioning

confidence: 99%

“…For the general development of implicative semilattice theory the ordered filters play an important role which is shown by Nemitz [7]. Motivated by this, Chan and Shum [3] established some elementary properties, and constructed quotient structure of implicative semigroups via ordered filters. Jun, Meng, and Xin [6] discussed ordered filters of implicative semigroups.…”

mentioning

confidence: 95%

Positive implicative ordered filters of implicative semigroups

Jun¹,

Kim²

2000

International Journal of Mathematics and Mathematical Sciences

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Abstract. We introduce the notion of positive implicative ordered filters in implicative semigroups. We show that every positive implicative ordered filter is both an ordered filter and an implicative ordered filter. We give examples that an ordered filter (an implicative ordered filter) may not be a positive implicative ordered filter. We also give equivalent conditions of positive implicative ordered filters. Finally we establish the extension property for positive implicative ordered filters.Keywords and phrases. Implicative semigroup, ordered filter, (positive) implicative ordered filter.2000 Mathematics Subject Classification. Primary 20M12, 06F05, 06A06, 06A12. Introduction.The notions of implicative semigroup and ordered filter were introduced by Chan and Shum [3]. The first is a generalization of implicative semilattice (see Nemitz [6] and Blyth [2]) and has a close relation with implication in mathematical logic and set theoretic difference (see Birkhoff [1] and Curry [4]). For the general development of implicative semilattice theory the ordered filters play an important role which is shown by Nemitz [7]. Motivated by this, Chan and Shum [3] established some elementary properties, and constructed quotient structure of implicative semigroups via ordered filters. Jun, Meng, and Xin [6] discussed ordered filters of implicative semigroups. Also Jun [6] stated implicative ordered filters of implicative semigroups. For a deep study of implicative semigroups it is undoubtedly necessary to establish more complete theory of ordered filters. In this paper, we introduce the notion of positive implicative ordered filters in implicative semigroups. We show that every positive implicative ordered filter is both an ordered filter and an implicative ordered filter. We give examples that an ordered filter (an implicative ordered filter) may not be a positive implicative ordered filter. We also give equivalent conditions of positive implicative ordered filters. Finally we establish the extension property for positive implicative ordered filters.We recall some definitions and results. By a negatively partially ordered semigroup (briefly, n.p.o. semigroup) we mean a set S with a partial ordering "≤" and a binary operation "·" such that for all x, y, z ∈ S, we have:(1) (x · y) · z = x · (y · z), (2) x ≤ y implies x · z ≤ y · z and z · x ≤ z · y, (3) x · y ≤ x and x · y ≤ y.

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