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In this paper, we study (open) filters and deductive systems of self-distributive weak Heyting algebras (SDWH-algebras) and obtain some results which determine the relationship between them. We show that the variety of SDWH-algebras is not weakly regular and every open filter is the kernel of at least one congruence relation. Finally, we characterize those SDWH-algebras which are weakly regular by using some properties involving principal congruence relations. [ABSTRACT FROM AUTHOR]

Background The deductive system of BCL-algebra has not been thoroughly explored, and its fuzzification remains undefined. This study aims to the association of this gap by introducing a fuzzy extension of the deductive system of BCL-algebra using Pythagorean fuzzy sets. Methods We define a Pythagorean fuzzy set as a pair of membership and non-membership functions, where the sum of their squares lies between 0 and 1. These functions are used to extend the classical deductive system of BCL-algebra, allowing for reasoning based on fuzzy degrees. The system incorporates fuzzy operations such as union, intersection, and complement, while maintaining the structure of BCL-algebra. We also introduce key components such as the accuracy function, score function, degree of indeterminacy, and square deviation to measure the certainty, truth, uncertainty, and deviation of fuzzy sets. Results We prove that the intersection of two Pythagorean fuzzy deductive systems remains a valid Pythagorean fuzzy deductive system within BCL-algebra. However, we show that the union of such systems does not necessarily form a valid fuzzy deductive system. The study also provides detailed proofs using induction, logical derivations, and algebraic techniques. Conclusion The results disclose that while the intersection of Pythagorean fuzzy deductive systems preserves the system's structure, the union does not, offering new insights into the behavior and limitations of fuzzy systems in BCL-algebra.

George Boole published the pamphlet The Mathematical Analysis of Logic in 1847. He believed that logic should belong to a universal mathematics that would cover both quantitative and nonquantitative research. With his pamphlet, Boole signalled an important change in symbolic logic: in contrast with his predecessors, his thinking was exclusively extensional. Notwithstanding the innovations introduced he accepted all traditional Aristotelean syllogisms. Nevertheless, some criticisms have been raised concerning Boole’s view of Aristotelean logic as the solution of algebraic equations. In order to circumvent such criticisms, we show here how Boole’s conclusions may be deduced from the axioms and inference rules proposed in his pamphlet, from which Aristotle’s deductions in Prior Analytics (including those that are completed through an impossibility) can be stated as theorems. We clarify his method for dealing with both universal and particular premises by means of his symbol v and demystify some criticisms concerning the way he expressed particular premises. Boole’s conclusions for some of those premises for which according to Aristotle there is no deduction are also discussed. The deductive system presented here has the potential to provide a new perspective on Boole’s approach to logic in The Mathematical Analysis of Logic. [ABSTRACT FROM AUTHOR]

To investigate the filter and deductive system of the Schaefer stroke Hilbert algebra using the Dokdo structure, the concept of Dokdo filter and Dokdo deductive system is defined, examples are given, and various properties are investigated. The Dokdo filter is formed by attaching appropriate conditions to the given Dokdo structure. Characterization of Dokdo filter is studied. Dokdo filters related to filters are constructed. Dokdo filter and Dokdo deductive system turn out to be the same concept. [ABSTRACT FROM AUTHOR]

We consider some generalizations of BCK algebras such as for example RM, tRM, aRM (= BH) algebras and others. We investigate modal operators on such algebras. In particular, we show that the composition of two modal operators on an aRM algebra verifying the transitivity property is also a modal operator if and only if they commute. It is also proved that if two modal operators on an aRM algebra have the same image, then they coincide. Moreover, we introduce and study modal RM algebras. We define modal upper sets of RM algebras and we investigate some properties of these sets. We construct quotient modal RM algebras and we prove that if φ is a homomorphism of a modal RM algebra A onto a modal aRM algebra B, then A/Ker(φ) is isomorphic to B. Furthermore, we establish connections between deductive systems and congruences for modal RM algebras. [ABSTRACT FROM AUTHOR]

We define and study deductive systems in unital quantum-B algebras. Then we introduce the notion of filter in a unital quantum-B algebras with product, and show that filters coincide with deductive systems. We also introduce and characterize the commutative deductive systems in unital quantum-B algebras. [ABSTRACT FROM AUTHOR]

Interval-valued neutrosophic sets (IVNSs) are a notion that was initially developed by Wang et al.19 The idea of IVNSs to deductive systems (DSs) in Hilbert algebras is presented in this study. It is shown how intervalvalued neutrosophic deductive systems (IVNDSs) relate to their level cuts. In addition, certain related features are examined as well as the homomorphic inverse image of IVNDSs in Hilbert algebras. [ABSTRACT FROM AUTHOR]

The notion of a RM algebra, introduced recently, is a generalization of many other algebras of logic. The class of RM algebras contains (weak-)BCC algebras, BCH algebras, BCI algebras, BCK algebras and many others. A RM algebra is an algebra A = (A; →, 1) of type (2, 0) satisfying the identities: x → x = 1 and 1 → x = x. In this paper we study the set of maximal elements of a RM algebra, branches of a RM algebra and moreover translation deductive systems of a RM algebra giving so called the Representation Theorem for RM algebras.

In this paper we introduce a new class of residuated lattice: residuated lattice with (C∧&→) property and we prove that (C∧&→) ⇔ (C→) + (C∧).Also, we introduce and characterize C→, C∨, C∧ and C∧ & → filters in residuated lattices (i.e., we characterize the filters for which the quotient algebra that is constructed via these filters is a residuated lattice with C→ (C∨ or C∧ or C∧&→ property). We state and prove some results which establish the relationships between these filters and other filters of residuated lattices: BL filters, MTL filters, divisible filters and, by some examples, we show that these filters are different. Starting from the results of algebras, we present for MTL filters, BL filters and C∧&→ filters the decomposition conditions.

Together with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that—similar to relatively pseudocomplemented lattices—these structures can serve as an algebraic semantics of certain intuitionistic logics. The aim of the present paper is to define congruences and filters in these structures, derive mutual relationships between them and describe basic properties of congruences in strongly sectionally pseudocomplemented posets. For the description of filters in both sectionally pseudocomplemented lattices and posets, we use the tools introduced by A. Ursini, i.e., ideal terms and the closedness with respect to them. It seems to be of some interest that a similar machinery can be applied also for strongly sectionally pseudocomplemented posets in spite of the fact that the corresponding ideal terms are not everywhere defined. [ABSTRACT FROM AUTHOR]

Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain "unsharp" residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some properties of homomorphisms and deductive systems of generalized BE-semigroups. In particular, we show that through every deductive system, we may get factor generalized BE-semigroup. [ABSTRACT FROM AUTHOR]

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* ⊙ y; x ⊙ y* 2 I, for x; y 2 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. [ABSTRACT FROM AUTHOR]

In this paper, we derive some new results on Noetherian and Boolean Artinian BL-algebras. We further obtain some relations between local and semilocal £¡L-algebras and Boolean Artinian BLalgebras. [ABSTRACT FROM AUTHOR]