Basic notions of (constructive) semigroups with apartness

Crvenković, Siniša; Mitrovic, Melanija; Romano, Daniel A.

doi:10.1007/s00233-016-9776-y

Cited by 15 publications

(38 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This investigation is in Bishop's constructive algebra in a sense of papers [10,11,22,23,24,23,28,29,30] and books [3,4,5,8], [31](Chapter 8: Algebra). Let (S, =, =) be a constructive set (i.e.…”

Section: Bishop's Constructive Orientationmentioning

confidence: 99%

“…For undefined notions and notations and used in this article, the reader can look in some of the following articles [10,11,22,23,24,25,26,27,28,29,30].…”

Section: Bishop's Constructive Orientationmentioning

confidence: 99%

“…The specificity of Intuitionistic logics make it possible to identify, understand and analyze the observed types of algebraic structures as specific relational systems. In the texts [10,11,22,23,25], the author alone or in collaboration with his associates analyzes semigroups with apartness. In such semigroups, among other things, the object of research is also the relations of coorder and co-quasiorder as logical dual(s) of order (and quasi-order relations) in semigroups.…”

Section: Final Observationmentioning

confidence: 99%

See 2 more Smart Citations

On Co-Filters of Implicative Semigroups With Apartness

2019

Acta Univ. Apulensis Math. Inform.

View full text Add to dashboard Cite

The setting of this research is the Bishop's constructive mathematics-a mathematics based on the Intuitionistic Logic and principled-philosophical orientation of Bishop's mathematics. Implicative semigroups with apartness introduced and analyzed by this author in his two recently published articles. In this paper, as a continuation of the research [26, 27], a description of co-filters was made in an implicative semigroup with apartness using one class of special subsets of such semi-group.

show abstract

Section: Bishop's Constructive Orientationmentioning

confidence: 99%

“…For undefined notions and notations and used in this article, the reader can look in some of the following articles [10,11,22,23,24,25,26,27,28,29,30].…”

Section: Bishop's Constructive Orientationmentioning

confidence: 99%

Section: Final Observationmentioning

confidence: 99%

See 1 more Smart Citation

On Co-Filters of Implicative Semigroups With Apartness

2019

Acta Univ. Apulensis Math. Inform.

View full text Add to dashboard Cite

show abstract

“…Fundamentals of constructive algebra can be found in [12] and in the dissertations [13,23]. Specific structures in the Bish frame are investigated in [3][4][5]16] (semigroups) and in [14,15] (rings, fields and modules), while a comprehensive overview about general algebraic structures can be found in [22].…”

Section: The Bishop's Constructive Casementioning

confidence: 99%

Perception of BCC-algebras under the Bishops principled-philosophical orientation: BCC-algebra with apartness

Romano¹

2019

Filomat

View full text Add to dashboard Cite

In this paper it is given a short introduction in a reconsideration about BCC-algebras under the light of Bishop's principled-philosophical orientation. At the first, it is introduced the concept of BCCalgebras into this orientation. Additionally, the consequences of the selected axioms in the determination of BCC-algebras with apartness are analyzed. Also, some substructures in the BCC-algebras with apartness that have no counterparts in the classical case and which appear as products of the chosen logical environment such as co-ideals are analyzed. At the end, two different results that can be viewed as the isomorphism theorem in for BCC-algebras are exposed. Definition 1.1 ([6]). By a BCC-algebra is a non-empty set X together with a binary internal operation ' •' and a distinguished element ' 0' such that the following axioms are satisfied: (BCC1) (∀x, y, z ∈ X)((y • z) • ((x • y) • (x • z)) = 0); (BCC2) (∀x ∈ X)(x • x = 0); (BCC3) (∀x ∈ X)(x • 0 = 0); (BCC4) (∀x ∈ X)(0 • x = x); and (BCC5) (∀x, y ∈ X)((x • y = 0 ∧ y • x = 0) =⇒ x = y). Remark 1.2. Let us observe that their form is coherent with the interpretation in which • stay for =⇒, 0 for and = for the semantical equivalence usually denoted by ≡.

show abstract

“…The concept of semigroups with apartness has been introduced in [7,8] as yet another application of Bishop's constructive mathematics [1,2] to algebraic structures [14,20]. In [3], Bridges and Reeves comment that "modern algebra has proved amenable to a [...] constructive treatment".…”

Section: Introductionmentioning

confidence: 99%

Inverse semigroups with apartness

Cherubini

Frigeri

2019

Semigroup Forum

View full text Add to dashboard Cite

The notion of semigroups with apartness has been introduced recently as a constructive counterpart of classical semigroups. On such structures, a constructive analogue of the isomorphism theorem has been proved, and quasiorder relations and related substructures have been studied. In this paper, we extend this approach by introducing inverse semigroups with apartness, a useful tool to describe partial symmetries in sets with apartness. We prove a constructive analogue of the isomorphism theorem for inverse semigroups and provide a characterisation of cocongruences on inverse semigroups.

show abstract

Basic notions of (constructive) semigroups with apartness

Abstract: We examine basic notions of special subsets and orders in the context of semigroups with apartness and prove constructive analogues of some classical theorems relating such subsets and orders.

Cited by 15 publications

References 11 publications

On Co-Filters of Implicative Semigroups With Apartness

On Co-Filters of Implicative Semigroups With Apartness

Perception of BCC-algebras under the Bishops principled-philosophical orientation: BCC-algebra with apartness

Inverse semigroups with apartness

Contact Info

Product

Resources

About