On pseudo residuated skew lattices
- PDF / 938,878 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 98 Downloads / 197 Views
ORIGINAL ARTICLE
On pseudo residuated skew lattices R. Koohnavard1 • A. Borumand Saeid1 Received: 30 March 2019 / Accepted: 25 March 2020 Ó Sociedad Matemática Mexicana 2020
Abstract In this paper, pseudo residuated skew lattices are defined as a non-commutative generalization of residuated skew lattices and their properties are investigated. It is shown that the class of all conormal pseudo residuated skew lattices forms a variety under some conditions. Dense, regular and strong elements are studied in a pseudo residuated skew lattice and the relationships between them are discussed. We define and investigate different classes of pseudo residuated skew lattices and show that any pseudo residuated skew chain with element 0 is local and also, any locally finite pseudo residuated skew lattice is local. It is shown that any strong pseudo residuated skew lattice is good and any locally finite pseudo residuated skew lattice is good under an extra condition too. Keywords (Local Locally finite Integral good Strong) pseudo residuated skew lattice Normal filter Dense Regular elements
Mathematics Subject Classification 03G10 06A75 06B20 03B75
1 Introduction Non-commutative lattices have been studied for over 65 years. To our knowledge, the first person to engage in their extended study was the physicist Jordan [10]. The study of skew lattices began with the 1989 paper of Leech [16]. Skew lattices are a generalization of lattices in which operations _; ^ are not commutative. In a skew lattice two different order concepts can be defined: the natural preorder, denoted by & A. Borumand Saeid [email protected] R. Koohnavard [email protected] 1
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
123
R. Koohnavard, A. B. Saeid
and the natural partial order denoted by , one weaker than the other. Though, unlike lattices, in general the admissible Hasse diagram representing the order structure of a skew lattice does not determine its algebraic structure [19]. Green’s relation D induced by the preorder is a congruence on ðA; _; ^Þ such that ðA=D; _; ^Þ is a lattice too. Borumand Saeid and Koohnavard defined residuated skew lattices as a non-commutative generalization of residuated lattices in which _; ^ are not commutative. They showed that Green’s relation D is a congruence on residuated skew lattice A and A=D is a residuated lattice. They proved that the class of all conormal residuated skew lattices forms a variety under an extra condition [2]. In this paper, we define pseudo residuated skew lattices as a non-commutative generalization of residuated skew lattices in which ; _; ^ are not-commutative. We want to generalize the notion of residuated skew lattices and define the operation as non-commutative. One of the main goals in this paper is to extend some results proved for residuated skew lattices to non-commutative residuated skew lattices. We investigate some properties of pseudo residuated skew lattices and the conne
Data Loading...
