The reverse order law of the (b, c)-inverse in semigroups

Chen, J.; Ke, Yuanyuan; Mosić, Dijana

doi:10.1007/s10474-016-0667-1

Cited by 13 publications

(5 citation statements)

References 22 publications

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“…Then AW X = A and XW W * = W * . As also 1 6 W * A = X then X ∈ R W * ∩L A and X = W −(W * ,A) . Pose Y = One can then check directly that the Moore-Penrose equations are satisfied.…”

Section: Aknowledgementsmentioning

confidence: 98%

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(b, c)-inverse, inverse along an element, and the Schützenberger category of a semigroup

Mary

2021

CGASA

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We prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schützenberger category of a semigroup. Applications to the Reverse Order Law are given. C Green's relations and the Schützenberger category of a semigroupIn this first section, we provide the reader with the necessary definitions and results regarding semigroups and categories. In particular, we recall the definition of the Schützenberger category of a semigroup and the interpretation of Green's relations in this setting. Section 2 then presents the main result of the article (Theorem C.7), that (b, c)-inverses (and inverses along an element) are genuine inverses when considered as morphisms in the corresponding Schützenberger category. Finally, applications to the Reverse Order Law are given in Section 3.

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“…Then AW X = A and XW W * = W * . As also 1 6 W * A = X then X ∈ R W * ∩L A and X = W −(W * ,A) . Pose Y = One can then check directly that the Moore-Penrose equations are satisfied.…”

Section: Aknowledgementsmentioning

confidence: 98%

“…C. 1 Green's preorders and relations In this article, S denotes a semigroup, E(S) its set of idempotents and S 1 the monoid generated by S. We recall below the definitions of Green's preorders and relations [5].…”

Section: Marymentioning

confidence: 99%

(b, c)-inverse, inverse along an element, and the Schützenberger category of a semigroup

Mary

2021

CGASA

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“…The above equality is known as the reverse order law of invertible elements. In general, the reverse order law does not hold for generalized inverses (see [1,11]). In this section, the reverse order laws for the hybrid (b, c)-inverse are obtained.…”

Section: Reverse Order Law For the Hybrid (B C)-inversementioning

confidence: 99%

Further results on hybrid (b,c)-inverses in rings

Wang

2019

Filomat

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Let R be a ring and b, c ∈ R. In this paper, the absorption law for the hybrid (b, c)-inverse in a ring is considered. Also, by using the Green's preorders and relations, we obtain the reverse order law of the hybrid (b, c)-inverse. As applications, we obtain the related results for the (b, c)-inverse. Definition 1.1. Let R be an associative ring and let b, c ∈ R. An element a ∈ R is (b, c)-invertible if there exists y ∈ R such that y ∈ (bRy) ∩ (yRc), yab = b, cay = c. If such y exists, it is unique and is denoted by a (b,c). Drazin [2] also presented an equivalent characterization for the (b, c)-inverse y of a as yay = y, yR = bR and Ry = Rc. As generalizations of (b, c)-inverses, hybrid (b, c)-inverses and annihilator (b, c)-inverses were introduced in [2]. The symbols lann(a) = { ∈ R : a = 0} and rann(a) = {h ∈ R : ah = 0} denote the sets of all left annihilators and right annihilators of a, respectively. Definition 1.2. Let a, b, c, y ∈ R. We say that y is a hybrid (b, c)-inverse of a if yay = y, yR = bR, rann(y) = rann(c). If such y exists, it is unique. In this article, we use the symbol a (b c) to denote the hybrid (b, c)-inverse of a. Definition 1.3. Let a, b, c, y ∈ R. We say that y is a annihilator (b, c)-inverse of a if yay = y, lann(y) = lann(b), lann(y) = lann(c).

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“…The equality (3.1) is known as the reverse order law of invertible elements. In general, the reverse order law does not hold for generalized inverses, for example, [6,7,18,25]. The following two lemmas will be useful in the sequel.…”

Section: Reverse Order Laws For the (B C)-inversementioning

confidence: 99%