A Categorial Equivalence for Semi-Nelson Algebras

We present a category equivalent to that of semi-Nelson algebras. The objects in this category are pairs consisting of a semi-Heyting algebra and one of its filters. The filters must contain all the dense elements of the semi-Heyting algebra and satisfy an additional technical condition. We also show that the category of dually hemimorphic semi-Nelson algebras is equivalent to that of dually hemimorphic semi-Heyting algebras.

Fuzzy Convergence, Fuzzy Neighborhood Convergence and I-Tolerance Structures for Groups

We introduce a category of fuzzy convergence groups, FCONVGRP a subcategory of the category of fuzzy convergence spaces, FCONV. Viewing [Formula: see text] as a complete Heyting algebra, we prove that the category of [Formula: see text]-tolerance groups, [Formula: see text]-TOLGRP is isomorphic to a subcategory of FCONVGRP. Since FCONV is a topological universe, and thereby possesses function space structure, upon invoking this, we are able, among others, to show that FCONVGRP is topological, and more importantly, it enables us to obtain a compatible fuzzy convergence function space structure on group of homeomorphisms. It is noticeable, however, that the category of fuzzy neighborhood convergence groups, FNCONVGRP — a supercategory of the well-known category FNS, of fuzzy neighborhood spaces, as well as the category of fuzzy neighborhood groups, FNGRP — a subcategory of FNCONVGRP exhibit nice relationships with FCONVGRP. It is important to note that the objects of FCONVGRP are homogeneous, this paves the way to present two pertinent characterization theorems on fuzzy convergence groups. Finally, introducing a category PSTOPGRP, of pseudotopological groups, we reveal the embeddings of FTOPGRP and PSTOPGRP into FCONVGRP.

Construction of some algebras of logics by using intuitionistic fuzzy filters on hoops

<abstract><p>In this paper, we define the notions of intuitionistic fuzzy filters and intuitionistic fuzzy implicative (positive implicative, fantastic) filters on hoops. Then we show that all intuitionistic fuzzy filters make a bounded distributive lattice. Also, by using intuitionistic fuzzy filters we introduce a relation on hoops and show that it is a congruence relation, then we prove that the algebraic structure made by it is a hoop. Finally, we investigate the conditions that quotient structure will be different algebras of logics such as Brouwerian semilattice, Heyting algebra and Wajesberg hoop.</p></abstract>

The lattices of 𝔏-fuzzy state filters in state residuated lattices

In the paper, we introduce 𝔏-fuzzy state filters in state residuated lattices and investigate their related properties, where 𝔏 is a complete Heyting algebra. Moreover, we study the 𝔏-fuzzy state co-annihilator of an 𝔏-fuzzy set with respect to an 𝔏-fuzzy state filter. Finally, using the 𝔏-fuzzy state co-annihilator, we investigate lattice structures of the set of some types of 𝔏-fuzzy state filters in state residuated lattices. In particular, we prove that: (1) the set FSF[L] of all 𝔏-fuzzy state filters is a complete Heyting algebra; (2) the set SνFSF[L] of all stable state filters relative to an 𝔏-fuzzy set ν is also a complete Heyting algebra; (3) the set IμFSF[L] of all involutory 𝔏-fuzzy state filters relative to an 𝔏-fuzzy state filter μ is a complete Boolean algebra.

Soft Set Theory Applied to Hoops

In this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ⊙ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it.

Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN+2 = fN, thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

Valid Arguments and Heyting Algebra using Multi Valued Logic

Over last three decades, multi valued logic (MVL) has been receiving considerable attention. So, we focus our concentration upon multi valued logic using some of the rules of mathematical logic, which can be used in developing artificial intelligence. Since Aristotle’s logic there were only two propositions. Later it was extended to n-valued logical proposition which is greater than 2, that is popularly known as multi valued logic proposition – they are true, false and unknowns. In this paper we will discuss about multi valued logic with 27- possible using Jaina logic and some of the rules as it gives the best results. In Jaina Logic, indeterminant means something which cannot describe more than one aspect at a time. So, we are going to consider each aspect separately and assign True or False. Then according to the given condition we can either apply min or max condition to get a precise solution.

A Residuated Lattice of L-Fuzzy Subalgebras of a Mono-Unary Algebra

Given a complete residuated lattice [Formula: see text] and a mono-unary algebra [Formula: see text], it is well known that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subsets of [Formula: see text] satisfy the same residuated lattice identities. In this paper, we show that [Formula: see text] and the residuated lattice [Formula: see text] of [Formula: see text]-fuzzy subalgebras of [Formula: see text] satisfy the same residuated lattice identities if and only if the Heyting algebra [Formula: see text] of subuniverses of [Formula: see text] is a Boolean algebra. We also show that [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) if and only if [Formula: see text] is a Boolean algebra (respectively, an [Formula: see text]-algebra) and [Formula: see text] is a Boolean algebra.