New congruence properties for Ramanujan’s $\phi $ function

2021 ◽  
pp. 1
Author(s):  
Ernest X. W. Xia
Keyword(s):  
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.

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.

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 On the Symmetries of theq-Bernoulli Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Taekyun Kim
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Reflection Symmetry ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sum ◽  
Congruence Properties

Kupershmidt and Tuenter have introduced reflection symmetries for theq-Bernoulli numbers and the Bernoulli polynomials in (2005), (2001), respectively. However, they have not dealt with congruence properties for these numbers entirely. Kupershmidt gave a quantization of the reflection symmetry for the classical Bernoulli polynomials. Tuenter derived a symmetry of power sum polynomials and the classical Bernoulli numbers. In this paper, we study the new symmetries of theq-Bernoulli numbers and polynomials, which are different from Kupershmidt's and Tuenter's results. By using our symmetries for theq-Bernoulli polynomials, we can obtain some interesting relationships betweenq-Bernoulli numbers and polynomials.

scholarly journals Congruence properties for a certain kind of partition functions

2016 ◽  
Vol 290 ◽  
pp. 739-772 ◽  
Author(s):  
Su-Ping Cui ◽  
Nancy S.S. Gu ◽  
Anthony X. Huang
Keyword(s):  
Partition Functions ◽  
Congruence Properties
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