Classification of unilinear residuated lattices

The aim of this paper is to extend results established by H. Ono and T. Kowalski regarding directly indecomposable commutative residuated lattices to the non-commutative case. The main theorem states that a residu- ated lattice A is directly indecomposable if and only if its Boolean center B(A) is {0,1}. We also prove that any linearly ordered residuated lattice and any local residuated lattice are directly indecomposable. We apply these results to prove some properties of the Boolean center of a residuated lattice and also define the algebras on subintervals of residuated lattices. It is known that the study of classical logic can be reduced to studying Boolean algebras. Therefore, the discussion of any type of non-classical logic raises a ques- tion about the corresponding abstract algebra. There has been much research in the field of fuzzy logic when the conjunction of the truth values structure is not necessarily commutative. Developing algebraic models for non-commutative multiple-valued logics is a central topic in the research of fuzzy systems and one such algebraic structure is the non-commutative residuated lattice. In the last few years a corresponding fuzzy theory has developed in parallel with the classical the- ory (2, 3, 19, 13). Commutative residuated lattices were first introduced by M. Ward and R.P. Dil- worth as a generalization of ideal lattices of rings. Recently, these structures have been studied in (10) and (18). Non-commutative residuated lattices, sometimes called pseudo-residuated lattices, biresiduated lattices or generalized residuated lat- tices, are the algebraic counterparts of substructural logics; i.e. logics which lack at least one of the three structural rules, namely contraction, weakening and exchange. Complete studies on non-commutative residuated lattices were developed in (1) and (17). The aim of this paper is to extend results proved by H. Ono and T. Kowalski for the case of commutative residuated lattices to the non-commutative case. The main theorem states that a residuated lattice A is directly indecomposable if and only if its Boolean center B(A) is {0,1}. We also prove that any local residuated lattice is directly indecomposable and derive some properties of the Boolean center of a residuated lattice. As an application of the Boolean center of a residuated lat- tice we prove that any subinterval (a,b) of a residuated lattice can be endowed with an algebraic structure of the same kind as the original one. These structures are

Bosbach and Rieă?an states on residuated lattices both are generalizations of probability measures on Boolean algebras. Just from the observation that both of them can be defined by using the canonical structure of the standard MV-algebra on the unit interval [0, 1], generalized Rieă?an states and two types of generalized Bosbach states on residuated lattices were recently introduced by Georgescu and Mure?an through replacing the standard MV-algebra with arbitrary residuated lattices as codomains. In the present paper, the Glivenko theorem is first extended to residuated lattices with a nucleus, which gives several necessary and sufficient conditions for the underlying nucleus to be a residuated lattice homomorphism. Then it is proved that every generalized Bosbach state (of type I, or of type II) compatible with the nucleus on a nucleus-based-Glivenko residuated lattice is uniquely determined by its restriction on the nucleus image of the underlying residuated lattice, and every relatively generalized Rieă?an state compatible with the double relative negation on an arbitrary residuated lattice is uniquely determined by its restriction on the double relative negation image of the residuated lattice. Our results indicate that many-valued probability theory compatible with nuclei on residuated lattices reduces in essence to probability theory on algebras of fixpoints of the underlying nuclei.

We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a sectionally residuated semilattice. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Lukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras (A, r, →, ⇝, 1) of type 〈3, 2, 2, 0〉 where (A, →, ⇝, 1) is a {→, ⇝, 1}-subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r(x, y, z) = (x · y) ∨ z. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras.

In this paper, by using the notion of upsets in residuated lattices and defining the operator Da (X), for an upset X of a residuated lattice L we construct a new topology denoted by τ a and (L, τ a ) becomes a topological space. We obtain some of the topological aspects of these structures such as connectivity and compactness. We study the properties of upsets in residuated lattices and we establish the relationship between them and filters. O. Zahiri and R. A. Borzooei studied upsets in the case of BL-algebras, their results become particular cases of our theory, many of them work in residuated lattices and for that we offer complete proofs. Moreover, we investigate some properties of the quotient topology on residuated lattices and some classes of semitopological residuated lattices. We give the relationship between two types of quotient topologies τ a/F and τ a − $\begin{array}{} \displaystyle \mathop {{\tau _a}}\limits^ - \end{array}$ . Finally, we study the uniform topology τ Λ ¯ $\begin{array}{} \displaystyle {\tau _{\bar \Lambda }} \end{array}$ and we obtain some conditions under which ( L / J , τ Λ ¯ ) $\begin{array}{} \displaystyle (L/J,{\tau _{\bar \Lambda }}) \end{array}$ is a Hausdorff space, a discrete space or a regular space ralative to the uniform topology. We discuss briefly the applications of our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

In this paper, we enlarge the language of non-commutative residuated lattices to provide a unified algebraic foundation for probabilities of fuzzy events in substructural logics, by adding an internal state that describes algebraic properties of states. The resulting class of algebras will be called non-commutative residuated lattices with internal states (or state residuated lattices for short). First, we prove that any perfect residuated lattice admits a nontrivial internal state and discuss some algebraic properties of internal states. Also, we give characterizations of divisible residuated lattices and idempotent residuated lattices, and obtain relationships between internal states and states on residuated lattices. Moreover, using some kinds of state filters, we present some characterizations of local state residuated lattices and their subclasses. Furthermore, we obtain that each local state commutative residuated lattice is either perfect or locally finite or peculiar. Finally, we prove that the class $SF[L]$ of all state filters in state residuated lattices is a complete Heyting algebra. In particular, by studying the state co-annihilator of a nonempty set with respect to a state filter, we prove that 1) the class $S_{X}SF[L]$ of all stable state filters relative a nonempty set $X$ in state residuated lattices is also a complete Heyting algebra, but it is not a subalgebra of the Heyting algebra $SF[L]$ ; 2) the class $I_{F}SF[L]$ of all involutory state filters relative a state filter $F$ in state residuated lattices is a complete Boolean algebra.

The main aim of this paper is to investigate the topologies that constructed by some ideals on residuated lattices and some topologies which induced by lattice ideals and distance functions on involutive residuated lattices. To begin with, we present that prime $\oplus $-ideals and prime $\boxplus $-ideals are coincident on $MTL$-algebras and give some new results about ideals on residuated lattices. In the following, we study an $i$-topology which is induced by an $i$-system on a residuated lattice $A$ and get that $A$ equipped with such an $i$-topology is a topological residuated lattice and give a characterization for such topological involutive residuated lattices. Meanwhile, we give a notion of $\mathcal {I}$-completion of a residuated lattice $A$ with respect to the $i$-topology induced by an $i$-system $\mathcal {I}$ and characterize the $\mathcal {I}$-completion of $A$ by means of the inverse limit of an inverse system. Finally, we show that the topology that induced by a lattice ideal and a distance function on an involutive residuated lattice is a semitopological residuated lattice and which coincides with some $i$-topological residuated lattice when the lattice ideal is an ideal of $A$.

In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in residuated lattice a new type of filter called special filter and investigate its properties. Finally, regular filter property in residuated lattices is introduced and is studied in details.

In this paper, a combination of algebraic and topological methods is applied to obtain new and structural results on Gelfand residuated lattices. It is demonstrated that Gelfand’s residuated lattices strongly tied up with the hull–kernel topology. Particularly, it is shown that a residuated lattice is Gelfand if and only if its prime spectrum, equipped with the hull–kernel topology, is normal. The class of soft residuated lattices is introduced, and it is shown that a residuated lattice is soft if and only if it is Gelfand and semisimple. Gelfand residuated lattices are characterized using the pure part of filters. The relation between pure filters and radicals in a Gelfand residuated lattice is described. It is shown that a residuated lattice is Gelfand if and only if its pure spectrum is homeomorphic to its usual maximal spectrum. The pure filters of a Gelfand residuated lattice are characterized. Finally, it is proved that a residuated lattice is Gelfand if and only if the hull–kernel and the \(\mathscr {D}\)-topology coincide on the set of maximal filters.

Generalized Bosbach and Riečan states based on relative negations in residuated lattices

We discuss fuzzy generalisations of information relations taking two classes of residuated lattices as basic algebraic structures. More precisely, we consider commutative and integral residuated lattices and extended residuated lattices defined by enriching the signature of residuated lattices by an antitone involution corresponding to the De Morgan negation. We show that some inadequacies in representation occur when residuated lattices are taken as a basis. These inadequacies, in turn, are avoided when an extended residuated lattice constitutes the basic structure. We also define several fuzzy information operators and show characterizations of some binary fuzzy relations using these operators.KeywordsInformation relationsInformation operatorsResiduated latticesFuzzy setsFuzzy logical connectives

"The notion of ideal in residuated lattices is introduced in [Kengne, P. C., Koguep, B. B., Akume, D. and Lele, C., L-fuzzy ideals of residuated lattices, Discuss. Math. Gen. Algebra Appl., 39 (2019), No. 2, 181–201] and [Liu, Y., Qin, Y., Qin, X. and Xu, Y., Ideals and fuzzy ideals in residuated lattices, Int. J. Math. Learn & Cyber., 8 (2017), 239–253] as a natural generalization of that of ideal in MV algebras (see [Cignoli, R., D’Ottaviano, I. M. L. and Mundici, D., Algebraic Foundations of many-valued Reasoning, Trends in Logic-Studia Logica Library 7, Dordrecht: Kluwer Academic Publishers, 2000] and [Chang, C. C., Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490]). If A is an MV algebra and I is an ideal on A then the binary relation x ∼I y iff x^{*}Ꙩ y; x Ꙩy^{*} ∈ I , for x; y ∈ A; is a congruence relation on A. In this paper we find classes of residuated lattices for which the relation ∼ I (defined for MV algebras) is a congruence relation and we give new characterizations for i-ideals and prime i-ideals in residuated lattices. As a generalization of the case of BL algebras (see [Lele, C. and Nganou, J. B., MV-algebras derived from ideals in BL-algebras, Fuzzy Sets and Systems, 218 (2013), 103–113]), we investigate the relationship between i-ideals and deductive systems in residuated lattices."

A comparative study of variable precision fuzzy rough sets based on residuated lattices

This paper mainly focus on building the ideals theory of non regular residuated lattices. Firstly, the notions of ideals and fuzzy ideals of a residuated lattice are introduced, their properties and equivalent characterizations are obtained; at the meantime, the relation between filter and ideal is discussed. Secondly, two types prime ideals of a residuated lattice are introduced, the relations between the two types ideals are studied, in some special residuated lattices (such as MTL-algebras, lattice implication algebras, BL-algebras), prime ideal and prime ideal of the second kind are coincide. At the meantime, the notions of fuzzy prime ideal and fuzzy prime ideal of the second kind on a residuated lattice are introduced, aiming at the relation between prime ideal and prime ideal of the second kind, we mainly investigate the fuzzy prime ideal of the second kind. Finally, we investigated the fuzzy congruence relations induced by fuzzy ideal, we construct a new residuated lattice induced by fuzzy congruences, the homomorphism theorem is given.

In this article, we put forward the concepts of nodal and conodal ideals in a residuated lattice and study some properties. We state some examples and theorems. We investigate the inverse image of a nodal (conodal) ideal under a homomorphism. In addition, we pay attention to the relationships with the other types of ideals and special sets in varieties of residuated lattices. At the same time, we give a characterization of nodal ideals in terms of congruences and we show that if L is an MTL-algebra and I is a non-principal nodal ideal, then L / I is a chain. We propose a characterization for Boolean residuated lattices ( L is a Boolean residuated lattice if and only if L is an involution semi-G-agebra) and we discuss briefly the applications of our results in varieties of residuated lattices. Finally, we introduce the concept of a fuzzy (nodal) ideal of a residuated lattice, and give some related results. After that we define the concept of fuzzy ideal of a residuated lattice with respect to a t-conorm briefly, S-fuzzy ideals and we prove Representation Theorem in residuated lattices.

The pseudo-linear semantics of interval-valued fuzzy logics