Filters theory on pseudo residuated skew lattices
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Filters theory on pseudo residuated skew lattices R. Koohnavard1 · A. Borumand Saeid1 Received: 1 March 2019 / Accepted: 27 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract In this paper, we focus on investigating some types of (skew) filters such as weak (positive) implicative (skew) filters on pseudo residuated skew lattices and obtain some properties of these (skew) filters. The relationship between these (skew) filters and other types of (skew) filters are discussed. It is shown that any weak positive implicative filter is a weak implicative and strong filter if it is a normal filter. Moreover, we characterize some types of filters by the properties of corresponding quotient algebras. Some equivalent characterizations for some filters are obtained too. Finally, a new characterization of these filters in a pseudo residuated skew lattice by a diagram is given. Keywords Pseudo residuated skew lattice · (Involution, obstinate, (weak) (positive) implicative) (skew) filter Mathematics Subject Classification 03G10 · 06A75 · 03G25
1 Introduction In general terms, a non-commutative lattice is an algebra, (A, ∨, ∧), where both ∨ and ∧ are associative, idempotent binary operations which are connected by laws of absorption. Pascual Jordan, motivated by questions in quantum logic, initiated the study of non-commutative lattices in 1949 paper [8]. Further developments in non-commutative lattice theory have occurred in a paper by Schein [15], and in two papers on near lattices by Schweigert [16] and [17] too. Green’s relation D induced by the preorder is a congruence on (A, ∨, ∧) such that (A/D, ∨, ∧) is a lattice too. Recently A. Borumand Saeid and R. Koohnavard defined residuated skew lattices as a non-commutative generalization of the residuated lattices that they were given by residuum on the skew lattices. They defined deductive system and skew deductive system in residuated skew lattices [2] and showed that deductive system and filter are equivalent, but skew deductive system and skew filter are not equivalent and any skew
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A. Borumand Saeid [email protected] R. Koohnavard [email protected]
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Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
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R. Koohnavard, A. Borumand Saeid
deductive system is a skew filter under some conditions [9]. Also, they defined pseudo residuated skew lattices and showed that the class of all conormal pseudo residuated skew lattices forms a variety under some conditions [11]. For better understanding of pseudo residuated skew lattices, we investigate filters and use them for classification and obtain pseudo residuated skew lattices structural properties. One of the main goals in this paper is to generalize some results for (skew) filters in residuated skew lattices to non-commutative residuated skew lattices. Another of our goals in this paper is to introduce other types of (skew) filters in the pseudo residuated skew lattices and
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