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Algebras of Logic deal with some algebraic structures, often bounded lattices, considered as models of certain logics, including logic as a domain of order theory. There are well known their importance and applications in social life to advance useful concepts, as for example computer algebra. Starting from results obtained by Di Nolla and Lettieri in [1], in which they analyzed the structure of finite BL-algebras, in this paper we find properties and give examples of commutative unitary rings R with its set of ideals Id (R) to be a BL-algebra of a given type. Moreover, we present properties of finite rings or rings with a finite number of ideals in their connections with BL-rings. [ABSTRACT FROM AUTHOR]

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra's temporal flow semantics for Hájek's basic logic, and Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Łukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform fashion, and extend them to infinitely-many other substructural logics. [ABSTRACT FROM AUTHOR]

naBL-algebras are non-associative generalizations of BL-algebras obtained from non-associative t-norms (nat-norms). In the present paper we propose a further generalization of BL-algebras where associativity is not required. Such generalization is based on a subclass of bivariate general overlap functions called inflationary. We call this non-associative generalization inflationary BL-algebras , and we discuss the main differences between the latter and the more specific class of inflationary BL-algebras. We show that the class n a BL of non-associative BL-algebras obtained from general overlap functions contains the class n a T of naBL-algebras obtained by nat-norms, and we provide a pictorial representation that summarizes these facts. We also prove some related properties, as well as a version of the well-known Chinese Remainder Theorem for these algebras but, under certain restrictions. Moreover, the notions of pseudo-automorphisms, automorphisms and their action on general overlap functions are used to obtain conjugated inflationary BL-algebras, as well as to obtain inflationary BL-algebras by distorting nat-norms by pseudo-automorphisms and, in the converse direction, to obtain naBL-algebras from inflationary BL-algebras via automorphisms. [ABSTRACT FROM AUTHOR]

Basic Logic BL, introduced by P. Hájek in 1998, is the logic of all continuous t-norms and their residua. The variety of BL-algebras forms the algebraic semantics of BL. Let L be a variety of BL-algebras, and let L (L) be its lattice of subvarieties, ordered by inclusion. L is called strictly join irreducible (SJI) if, whenever L is the join of a set S of varieties of BL-algebras, then L ∈ S. Every variety in L (L) is obtained as join of SJI varieties, which may be considered as the building blocks of all the varieties in L (L). In a previous work by the second author, a partial classification for the SJI varieties of BL-algebras has been found. In this paper we provide a full classification of the SJI varieties of BL-algebras. Our main result is that a variety of BL-algebras is SJI iff it is generated by a BL-chain with finitely many components, each of them being a cancellative hoop or a Wajsberg hoop with finite rank. As an application, we provide a characterization for the varieties of BL-algebras having only finitely many subvarieties, and we study some additional topics. [ABSTRACT FROM AUTHOR]

In this paper, we present a method to describe (functionally) free algebras in some subvarieties of BL-algebras. Particularly, we give a description of free algebras in subvarieties of the subvariety MG , where MG is the subvariety of BL generated by the algebra [ 0 , 1 ] MV ⊕ [ 0 , 1 ] G. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for a BL-algebra to be isomorphic to a poset product of BL-chains which are poset product indecomposable. [ABSTRACT FROM AUTHOR]

In this paper we investigate splitting algebras in varieties of logics, with special consideration for varieties of BL-algebras and similar structures. In the case of the variety of all BL-algebras a complete characterization of the splitting algebras is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we defined the prime-like filters of B L -algebras, and proved some relations between this filter and other types of filters in B L -algebras. Also, we found the relation between prime-like filters and constructed the quotient B L -algebras via this filter. We proved some results that have proven for prime filters of B L -algebras, to the case of prime-like filters of B L -algebras. In this way, we found new properties for the primary filters of B L -algebras. Also, we introduced the notions of prime-like B L -algebras (P l − B L -algebras), and Top-like B L -algebras. And we applied the notion of prime-like filters to define and study a new topology on B L -algebras. [ABSTRACT FROM AUTHOR]

In this paper, we use the generalize d rotation construction to lift results from the lattice of subvarieties of basic hoops to some parts of the lattice of subvarieties of monoidal t-norm based logic-algebras. In particular, we study splitting algebras for (the lattice of subvarieties of) varieties generated by generalized rotations of basic hoops and relevant subvarieties such as Wajsberg hoops, cancellative hoops and Gödel hoops. Finally, we show that the generalized rotation construction preserves the amalgamation property. [ABSTRACT FROM AUTHOR]

Fuzzy possibilistic logic is an important formalism for approximate reasoning. It extends the well-known basic propositional logic BL, introduced by Hájek, by offering the ability to reason about possibility and necessity of fuzzy propositions. We consider an algebraic approach to study this logic, introducing Pseudomonadic BL-algebras. These algebras turn to be a generalization of both Pseudomonadic algebras introduced by Bezhanishvili (Math Log Q 48:624-636, 2002) and serial, Euclidean and transitive Bimodal Gödel algebras proposed by Caicedo and Rodriguez (J Log Comput 25:37-55, 2015). We present the connection between this class of algebras and possibilistic BL-frames, as a first step to solve an open problem proposed by Hájek (Metamathematics of fuzzy logic. Trends in logic, Kluwer, Dordrecht, 1998, Chap. 8, Sect. 3). [ABSTRACT FROM AUTHOR]

Different constructions of BL-chains are compared. We establish when the ordinal sum and the poset product of the same family of BL-chains coincide. We also compare the poset product of MV-chains and product chains with saturated BL-chains. [ABSTRACT FROM AUTHOR]

In this paper we introduce a poset of subvarieties of BL-algebras, whose completion is the entire lattice of subvarietes; we exhibit also a description of this poset in terms of finite sequences of functions on the natural numbers. [ABSTRACT FROM AUTHOR]

In this article we introduce the variety of monadic BL-algebras as BL-algebras endowed with two monadic operators ∀ and ∃. After a study of the basic properties of this variety we show that this class is the equivalent algebraic semantics of the monadic fragment of Hájek's basic predicate logic. In addition, we start a systematic study of the main subvarieties of monadic BL-algebras, some of which constitute the algebraic semantics of well-known monadic logics: monadic Gödel logic and monadic Łukasiewicz logic. In the last section we give a complete characterization of totally ordered monadic BL-algebras. [ABSTRACT FROM AUTHOR]

A representation theorem for a class of BL-algebras with finite spectrum is presented. Although the class comprised by our result is not the whole class of BL-algebras with finite spectrum, it applies to some important classes such as finite BL-algebras and BL-chains with finite spectrum among others. Our representation constitutes a generalization of the ordinal sum construction, since we decompose each algebra in terms of totally ordered Wajsberg hoops. [ABSTRACT FROM AUTHOR]

We introduce an expansion of Basic Logic (BL) with new connectives which express fixed points of continuous formulas, i.e. formulas of BL whose connectives are among $$ \{ \& ,\vee ,\wedge \}$$ . The algebraic semantics of this logic is studied together with some of its subclasses corresponding to extensions of the above-mentioned expansion. The axiomatic extensions are proved to be standard complete. [ABSTRACT FROM AUTHOR]

In BL-algebras we introduce the concept of generalized co-annihilators as a generalization of coannihilator and the set of the form x-1F where F is a filter, and study basic properties of generalized co-annihilators. We also introduce the notion of involutory filters relative to a filter F and prove that the set of all involutory filters relative to a filter with respect to the suit operations is a complete Boolean lattice and BL-algebra. We use the technology of generalized co-annihilators to give characterizations of prime filters and minimal prime filters, respectively. In particular, we give a representation of co-annihilators in the quotient algebra of a BL-algebra L via a filter F by means of generalized co-annihilators relative to F in L:

In this paper, we shall give answers to show that two open problems, which is proposed by Turunen et al. in (Erratum to :) n-Fold implicative basic logic is Gödel logic [Soft Computing 16(2012): 177-181,183]. That is, each n-fold positive implicative filter is an n-fold Boolean filter and in an n-fold fantastic BL-algebra, n-fold implicative basic logic and n-fold positive implicative basic logic coincide. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of nodes and nodal filters in BL-algebras. Then we study the relationships between nodal filters and some other types of filters, likeness prime filter and special filter in BL-algebras. Also, we study inverse image of a nodal filter under a BL-homomorphism. Finally, we obtain a characterization of the nodal filters in terms of congruences. [ABSTRACT FROM AUTHOR]

In this paper, the intuitionistic fuzzy filter theory of BL-algebras is researched. The basic knowledge of BL-algebras and intuitionistic fuzzy sets is firstly reviewed. The notions of intuitionistic fuzzy filters, lattice filters, prime filters, Boolean filters, implicative filters, positive implicative filters, ultra filters and obstinate filters are introduced, respectively. Their important properties are investigated. In intuitionistic fuzzy sets, intuitionistic fuzzy filters, Boolean filters, ultra filters are proved to be equivalent to lattice filters, implicative filters, obstinate filters, respectively. Each intuitionistic fuzzy Boolean filter is an intuitionistic fuzzy positive implicative filter, but the converse may not be true in BL-algebras. The conditions under an intuitionistic fuzzy positive implicative filter being an intuitionistic fuzzy Boolean filter are constructed. Finally, the concepts of the intuitionistic fuzzy ultra and obstinate filters are introduced, and the intuitionistic fuzzy ultra filter is proved to be equivalent to the intuitionistic fuzzy obstinate filter in BL-algebras. [ABSTRACT FROM AUTHOR]

We present the logic BL, an axiomatic extension of BL (see []) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BL-algebras will be strictly connected to the one generated by Chang's MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some new results concerning these last structures and their logic. [ABSTRACT FROM AUTHOR]

We further study the fantastic filter and normal filter of BL-algebras. By studying the equivalent condition of fantastic filter, we reveal the relation between fantastic filter and normal filter of BL-algebras and we solve two open problems that "Under what suitable condition a normal filter becomes a fantastic filter?" and "(Extension property for a normal filter) Under what suitable condition extension property for normal filter holds?". [ABSTRACT FROM AUTHOR]

The spectrum of a residuated lattice L is the set Spec( L) of all prime i-filters. It is well known that Spec( L) can be endowed with the spectral topology. The main scope of this paper is to introduce and study another topology on Spec( L), the so called stable topology, which turns out to be coarser than the spectral one. With this and in view, we introduce the notions of pure i-filter for a residuated lattice and the notion of normal residuated lattice. So, we generalize to case of residuated lattice some results relative to MV-algebras (Belluce and Sessa in Quaest Math 23:269-277, ; Cavaccini et al. in Math Japonica 45(2):303-310, ) or BL-algebras (Eslami and Haghani in Kybernetika 45:491-506, ; Leustean in Central Eur J Math 1(3): 382-397, ; Turunen and Sessa in Mult-Valued Log 6(1-2):229-249, ). [ABSTRACT FROM AUTHOR]

In the present paper we extend the results of [4] by completely characterizing dual canonical subvarieties of BL-algebras. These are subvarieties of algebras that satisfy the equation xk = xk+1 for some integer k ≥ 1. As a corollary we get a full description of subvarieties of BL-algebras that admit completions. [ABSTRACT FROM AUTHOR]

We prove that Haveshki's and Eslami's n-fold implicative basic logic is Gödel logic and n-fold positive implicative basic logic is a fragment of ukasiewicz logic. [ABSTRACT FROM AUTHOR]

Abstract: For each integer , we provide an explicit functional characterization of the free -generated BL-algebra, together with an explicit construction of the corresponding normal forms. [Copyright &y& Elsevier]

We prove that every subvariety of BL-algebras which is not finitely generated is not σ-canonical. We also prove π-canonicity for an infinite family of subvarieties of BL-algebras that are not finitely generated. To do so we study the behavior of canonical extensions of ordered sums of posets. [ABSTRACT FROM AUTHOR]

In this paper we define Baer BL-algebras as BL-algebras with the property that co-annihilator filters are generated by central elements. We use sheaf-theoretic techniques to construct a Baer extension of any BL-algebra, that is to embed any nontrivial BL-algebra A into a Baer BL-algebra A*. The embedding turns to be an isomorphism if A is itself a Baer BL-algebra. [ABSTRACT FROM AUTHOR]

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces. [ABSTRACT FROM AUTHOR]

In this paper we study the prime and maximal spectra of a BL-algebra, proving that the prime spectrum is a compact T 0 topological space and that the maximal spectrum is a compact Hausdorff topological space. We also define and study the reticulation of a BL-algebra. [ABSTRACT FROM AUTHOR]

Pseudo-BL-algebras were introduced by di Nola, Georgescu and Iorgulescu as a non-commutative generalization of BL-algebras. BL-algebras serve as general semantics of the basic fuzzy logic BL. Here we develop a corresponding generalization psBL (pseudo-basic fuzzy logic) of the logic BL and show the relation of psBL to pseudo-BL-algebras. [ABSTRACT FROM PUBLISHER]

In this paper we investigate the falsehood-free fragments of main residuated fuzzy logics related to continuous t-norms (Hájek's Basic fuzzy logic BL and some well-known axiomatic extensions), and we relate them to the varieties of 0-free subreducts of the corresponding algebras. These turn out to be classes of algebraic structures known as hoops. We provide axiomatizations of all these fragments and we call them hoop logics; we prove they are strongly complete with respect to their corresponding classes of hoops, and that each fuzzy logic is a conservative extension of the corresponding hoop logic. Analogously, we also study the falsehood-free fragment of a weaker logic than BL, called MTL, which is the logic of left-continuous t-norms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsehood-free fragments of the fuzzy predicate calculi of the above logics and show completeness and conservativeness results. The role of axiom (∀3) in these predicate logics is studied. Finally, computational complexity issues of the propositional logics are also addressed. [ABSTRACT FROM PUBLISHER]

Consider any subvariety of BL-algebras generated by a single BL-chain which is the ordinal sum of the standard MV-algebra on [ 0 , 1 ] and a basic hoop H. We present a geometrical characterization of elements in the finitely generated free algebra of each of these subvarieties. In this characterization there is a clear insight of the role of the regular and dense elements of the generating chain. As an application, we analyze maximal and prime filters in the free algebra. [ABSTRACT FROM AUTHOR]