Abstract
In this article, we continue our research on quasi-ordered residuated systems introduced in 2018 by S. Bonzio and I. Chajda and various types of filters in them. Some fundamental properties of strong quasi-ordered residuated systems are given in this article. In addition, the concepts of prime and irreducible filters in such systems are introduced and analyzed.
Highlights
The concept of residuated relational systems ordered under a quasi-order relation, or quasi-ordered residuated systems, was introduced in 2018 by S
We have developed the concepts of implicative, associated, comparative and weak implicative filters in quasi-ordered residuated systems: Definition 4 ([5])
Definition 7. ([8]) For a non-empty subset F of a quasi-ordered residuated system A we say that the weak implicative filter in A if (F2) and the following condition (WIF) (∀u, v, z ∈ A)((u → (v → z) ∈ F ∧ u → v ∈ F) =⇒ u → (u → z) ∈ F)
Summary
The concept of residuated relational systems ordered under a quasi-order relation, or quasi-ordered residuated systems (briefly, QRS), was introduced in 2018 by S. The concept of a strong quasi-ordered residuated system was introduced and discussed in [9]. In such systems, comparative and implicative filters coincide. The concepts of prime (Definition 9) and irreducible (Definition 10) filters in strong quasi-ordered residuated systems will be introduced and some their important properties will be recognized (Theorem 5, Theorem 6 and Theorem 7). It is shown (Theorem 9) that any prime filter in a strong quasi-ordered residuated system is an irreducible filter
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