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Weak implicative filters in quasi-ordered residuated systems. (English) Zbl 1521.08001

Summary: The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda [Asian-Eur. J. Math. 11, No. 2, Article ID 1850024, 14 p. (2018; Zbl 1403.08001)] as a structure \(A = \langle A, \cdot,\rightarrow, 1, R \rangle\), where \((A, \cdot)\) 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.

MSC:

08A02 Relational systems, laws of composition
06A11 Algebraic aspects of posets
06B75 Generalizations of lattices

Citations:

Zbl 1403.08001
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References:

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