Congruence properties of pseudocomplemented De Morgan algebras

2014 ◽  
Vol 60 (6) ◽  
pp. 425-436 ◽  
Author(s):  
Hanamantagouda P. Sankappanavar ◽  
Júlia Vaz de Carvalho
Keyword(s):  
De Morgan Algebras ◽  
Congruence Properties

scholarly journals Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 753
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Universal Algebra ◽  
Relational Structures ◽  
Pseudocomplemented Poset ◽  
The Given ◽  
The Relationship ◽  
Congruence Properties

In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.

Symmetric operators on modal pseudocomplemented De Morgan algebras

2017 ◽  
Vol 25 (4) ◽  
pp. 496-511
Author(s):  
Aldo Figallo-Orellano ◽  
Alicia Ziliani ◽  
Martín Figallo
Keyword(s):  
De Morgan Algebras ◽  
Symmetric Operators

Weakly orthomodular and dually weakly orthomodular posets

2018 ◽  
Vol 11 (02) ◽  
pp. 1850093 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Orthomodular Poset ◽  
Algebraic Semantic ◽  
Orthomodular Posets ◽  
Congruence Properties

Orthomodular posets form an algebraic semantic for the logic of quantum mechanics. We show several methods how to construct orthomodular posets via a representation within the powerset of a given set. Further, we generalize this concept to the concept of weakly orthomodular and dually weakly orthomodular posets where the complementation need not be antitone or an involution. We show several interesting examples of such posets and prove which intervals of these posets are weakly orthomodular or dually weakly orthomodular again. To every (dually) weakly orthomodular poset can be assigned an algebra with total operations, a so-called (dually) weakly orthomodular [Formula: see text]-lattice. We study properties of these [Formula: see text]-lattices and show that the variety of these [Formula: see text]-lattices has nice congruence properties.

scholarly journals Distributive p-algebras and double p-algebras having n-permutable congruences

1992 ◽  
Vol 35 (2) ◽  
pp. 301-307 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Order Theory ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
First Order ◽  
First Order Theory ◽  
De Morgan Algebras ◽  
Element Chain

Recent research on aspects of distributive lattices, p-algebras, double p-algebras and de-Morgan algebras (see [2] and the references therein) has led to the consideration of the classes (n≧1) of distributive lattices having no n + 1-element chain in their poset of prime ideals. In [1] we were obliged to characterize the members of by a sentence in the first-order theory of distributive lattices. Subsequently (see [2]), it was realised that coincides with the class of distributive lattices having n+1-permutable congruences. This result is hereby employed to describe those distributive p-algebras and double p-algebras having n-permutable congruences. As an application, new characterizations of those distributive p-algebras and double p-algebras having the property that their compact congruences are principal are obtained. In addition, those varieties of distributive p-algebras and double p-algebras having n-permutable congruences are announced.

scholarly journals The ideal lattice of an MS-algebra

1988 ◽  
Vol 30 (2) ◽  
pp. 137-143 ◽  
Author(s):  
T. S. Blyth ◽  
J. C. Varlet
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Unary Operation ◽  
Stone Algebra ◽  
Ideal Lattice ◽  
Prime Filter ◽  
De Morgan Algebra ◽  
De Morgan Algebras ◽  
The Ideal

Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

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