Weak implicative filters in quasi-ordered residuated systems





Quasi-ordered residuated relational system, Filter, Implicative filter, Weak implicative filter


The concept of residuated relational systems ordered under a quasiorder relation was introduced in 2018 by S. Bonzio and I. Chajda as a structure A = 〈A, ·,→, 1, R〉, where (A, ·) 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.


S. Bonzio, "Algebraic structures from quantum and fuzzy logics", PhD Thesis, Università degli studi di Cagliari, 2016. [On line]. Available: https://bit.ly/3bUmb5N

S. Bonzio and I. Chajda, “Residuated relational systems”, Asian-European journal of mathematics, vol. 11, no. 2, Art ID. 1850024, 2018, doi: 10.1142/S1793557118500249

B. Jónsson and A. Tarski, “Representation problems for relation algebras” [Abstract], in Bulletin of the American Mathematical Society, 1948, vol. 54, no. 1, p. 80, doi: 10.1090/S0002-9904-1948-08948-0

R. D. Maddux, “The origin of relation algebras in the development and axiomatization of the calculus of relations”, Studia logica, vol. 50, no. 3, pp. 421-455, 1991, doi: 10.1007/BF00370681

D. A. Romano, “Filters in residuated relational system ordered under quasi-order”, Bulletin International Mathematics Virtual Institute, vol. 10, no. 3, pp. 529-534, 2020, doi: 10.7251/BIMVI2003529R

D. A. Romano, “Implicative filters in quasi-ordered residuated system”, Proyecciones (Antofagasta), vol. 40, no. 2, pp. 417-424, 2021, doi: 10.22199/issn.0717-6279-2021-02-0025

A. Tarski, “On the calculus of relations”, The journal of symbolic logic, vol. 6, no. 3, pp. 73-89, 1941, doi: 10.2307/2268577

A. Tarski and S. R. Givant, A formalization of set theory without variables. Providence, RI: AMS, 1987.



How to Cite

D. Romano, “Weak implicative filters in quasi-ordered residuated systems”, Proyecciones (Antofagasta, On line), vol. 40, no. 3, pp. 797-804, May 2021.