scholarly journals Implicative filters in quasi-ordered residuated systems

2021 ◽  
Vol 40 (2) ◽  
pp. 481-504
Author(s):  
Daniel A. Romano
Keyword(s):  
Algebraic Structures ◽  
Commutative Monoid ◽  
Ordered Monoid ◽  
Relational Systems ◽  
Quasi Order

The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure 𝒜 = 〈A, ·,→, 1, R〉, where (A, ·) is a commutative monoid with the identity 1 as the top element in this ordered monoid under a quasi-order R. The author introduced and analyzed the concepts of filters in this type of algebraic structures. In this article, as a continuation of previous author’s research, the author introduced and analyzed the concept of implicative filters in quasi-ordered residuated systems.

scholarly journals Weak implicative filters in quasi-ordered residuated systems

2021 ◽  
Vol 40 (3) ◽  
pp. 797-804
Author(s):  
Daniel A. Romano
Keyword(s):  
Algebraic Structures ◽  
Commutative Monoid ◽  
Ordered Monoid ◽  
Relational Systems ◽  
Quasi Order

The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure A = 〈A, ·,→, 1, R〉, where (A, ·) is a commutative monoid with the identity 1 as the top element in this ordered monoid under a quasi-order R. The author introduced and analyzed the concepts of filters and implicative filters in this type of algebraic structures. In this article, the concept of weak implicative filters in a quasi-ordered residuated system is introduced as a continuation of previous researches. Also, some conditions for a filter of such system to be a weak implicative filter are listed.

scholarly journals Strong quasi-ordered residuated system

2021 ◽  
Vol 5 (1) ◽  
pp. 73-79
Author(s):  
Daniel A. Romano ◽  
Keyword(s):  
Upper Bound ◽  
Order Relation ◽  
Algebraic Systems ◽  
Relational Systems ◽  
Quasi Order

The concept of residuated relational systems ordered under a quasi-order relation was introduced in 2018 by S. Bonzio and I. Chajda. In such algebraic systems, we have introduced and developed the concepts of implicative and comparative filters. In addition, we have shown that every comparative filter is an implicative filter at the same time and that converse it does not have to be. In this article, as a continuation of previous research, we introduce the concept of strong quasi-ordered residuated systems and we show that in such systems implicative and comparative filters coincide. In addition, we show that in such systems the concept of least upper bound for any two pair of elements can be determined.

scholarly journals S-acts over a Well-ordered Monoid with Modular Congruence Lattice

2021 ◽  
Vol 35 ◽  
pp. 87-102
Author(s):  
A.A. Stepanova ◽  
Keyword(s):  
Universal Algebra ◽  
Congruence Lattice ◽  
Binary Operation ◽  
Ordered Set ◽  
Minimal Element ◽  
Structural Theory ◽  
Commutative Monoid ◽  
Ordered Monoid ◽  
Linearly Ordered Set

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$–act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$–acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$–acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.

Tensor products of polyadic algebras

1963 ◽  
Vol 28 (3) ◽  
pp. 177-200 ◽  
Author(s):  
Aubert Daigneault
Keyword(s):  
Boolean Algebra ◽  
Algebraic Logic ◽  
Algebraic Structures ◽  
Locally Finite ◽  
Elementary Equivalence ◽  
First Order ◽  
Polyadic Algebras ◽  
Logical Concept ◽  
Algebraic Treatment ◽  
Relational Systems

A basic concept of the theory of models is that of elementary equivalence of similar relational systems: two such systems are said to be elementarily equivalent if they satisfy the same first-order statements or, in other words, if they have the same (first-order) complete theory. It is possible to reformulate this notion of elementary equivalence of systems within the framework of algebraic logic by replacing theories by algebraic structures derived from them or more directly from the systems which are models of these theories. To any such theory T (or model of it), is indeed associated a locally finite polyadic algebra with equality, the underlying Boolean algebra of which is simply the well-known Tarski-Lindenbaum algebra of the theory. It is not hard to prove (see Section 6.1) that two systems are elementarily equivalent iff (i.e. if and only if) they have isomorphic polyadic. algebras. The possibility of replacing theories by algebraic structures and of reducing the purely logical concept of elementary equivalence to the algebraic one of isomorphism can be exploited to give a purely algebraic treatment of model-theoretic problems and suggests natural questions concerning these structures. The present paper illustrates that possibility.

On a product of universal relational systems

2020 ◽  
Vol Accepted ◽  
Author(s):  
Nitima Phrommarat ◽  
Sivaree Sudsanit
Keyword(s):  
Relational Systems

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.

Using the co-existence approach to achieve combined functionality of object-oriented and relational systems

1993 ◽  
Vol 22 (2) ◽  
pp. 109-118 ◽  
Author(s):  
R. Ananthanarayanan ◽  
V. Gottemukkala ◽  
W. Kaefer ◽  
T. J. Lehman ◽  
H. Pirahesh
Keyword(s):  
Object Oriented ◽  
Relational Systems
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