scholarly journals Constructing Some Logical Algebras with Hoops

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1243
Author(s):  
M. Aaly Kologani ◽  
Seok-Zun Song ◽  
R. A. Borzooei ◽  
Young Bae Jun
Keyword(s):  
Congruence Relation ◽  
Heyting Algebras ◽  
Algebraic Structures ◽  
Hilbert Algebras ◽  
Wajsberg Hoops

In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion of co-filter on hoops is introduced and related properties are investigated. Then by using of co-filter, a congruence relation on hoops is defined, and the associated quotient structure is studied. Thus Brouwerian semilattices, Heyting algebras, Wajsberg hoops, Hilbert algebras and BL-algebras are obtained.

scholarly journals Quantum B-algebras with involutions

2020 ◽  
pp. 2150233
Author(s):  
Lavinia Corina Ciungu
Keyword(s):  
Algebraic Structures ◽  
Wajsberg Hoops ◽  
Residuated Poset ◽  
One To One ◽  
Universal Quantifiers ◽  
The Relationship ◽  
Mv Algebras

The aim of this paper is to define and study the involutive and weakly involutive quantum B-algebras. We prove that any weakly involutive quantum B-algebra is a residuated poset. As an application, we introduce and investigate the notions of existential and universal quantifiers on involutive quantum B-algebras. It is proved that there is a one-to-one correspondence between the quantifiers on weakly involutive quantum B-algebras. One of the main results consists of proving that any pair of quantifiers is a monadic operator on weakly involutive quantum B-algebras. We investigate the relationship between quantifiers on bounded sup-commutative pseudo BCK-algebras and quantifiers on other related algebraic structures, such as pseudo MV-algebras and bounded Wajsberg hoops.

scholarly journals Sheffer Stroke Nelson Algebras

2021 ◽  
Author(s):  
Tahsin Oner ◽  
Tugce Katican ◽  
Arsham Borumand Saeid
Keyword(s):  
Modular Lattice ◽  
Congruence Relation ◽  
Algebraic Structures ◽  
Sheffer Stroke ◽  
Nelson Algebra ◽  
The Family ◽  
Nelson Algebras

Abstract In this study, Sheffer stroke Nelson algebras (briefly, s-Nelson algebras), (ultra) ideals, quasi-subalgebras and quotient sets on these algebraic structures are introduced. The relationships between s-Nelson and Nelson algebras are analyzed. Also, it is shown that a s-Nelson algbera is a bounded distributive modular lattice, and the family of all ideals forms a complete distributive modular lattice. A congruence relation on s-Nelson algebra is determined by its ideal and quotient s-Nelson algebras are constructed by this congruence relation. Finally, it is indicated that a quotient s-Nelson algebra defined by the ultra ideal is totally ordered and that the cardinality of the quotient is less than or equals to 2.

scholarly journals Residuated Structures and Orthomodular Lattices

Studia Logica ◽  
2021 ◽  
Author(s):  
D. Fazio ◽  
A. Ledda ◽  
F. Paoli
Keyword(s):  
Algebraic Logic ◽  
Residuated Lattices ◽  
Quantum Structures ◽  
Heyting Algebras ◽  
Quantum Algebras ◽  
Orthomodular Lattices ◽  
Lattice Of Subvarieties ◽  
Residuated Structures ◽  
Common Framework ◽  
Mv Algebras

AbstractThe variety of (pointed) residuated lattices includes a vast proportion of the classes of algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among the outliers, one counts orthomodular lattices and other varieties of quantum algebras. We suggest a common framework—pointed left-residuated $$\ell $$ ℓ -groupoids—where residuated structures and quantum structures can all be accommodated. We investigate the lattice of subvarieties of pointed left-residuated $$\ell $$ ℓ -groupoids, their ideals, and develop a theory of left nuclei. Finally, we extend some parts of the theory of join-completions of residuated $$\ell $$ ℓ -groupoids to the left-residuated case, giving a new proof of MacLaren’s theorem for orthomodular lattices.

Some Fixpoint Techniques in Algebraic Structures and Applications to Computer Science

1987 ◽  
Vol 10 (4) ◽  
pp. 387-413
Author(s):  
Irène Guessarian
Keyword(s):  
Logic Programming ◽  
Computer Science ◽  
Proof Theory ◽  
Algebraic Structures ◽  
Data Bases

This paper recalls some fixpoint theorems in ordered algebraic structures and surveys some ways in which these theorems are applied in computer science. We describe via examples three main types of applications: in semantics and proof theory, in logic programming and in deductive data bases.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

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