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In this paper, we introduce the notions of connexive and bi-intuitionistic multilattices and develop on their base the algebraic semantics for Kamide, Shramko, and Wansing's connexive and bi-intuitionistic multilattice logics which were previously known in the form of sequent calculi and Kripke semantics. We prove that these logics are sound and complete with respect to the presented algebraic structures. [ABSTRACT FROM AUTHOR]

This paper presents a novel concept of a polyatomic logic and initiates its systematic study. This approach, inspired by inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations , through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalized Blok–Esakia theory, which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of polyatomic logics and the general Blok–Esakia theory for a wide class of interesting translations. [ABSTRACT FROM AUTHOR]

We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let Ord denote the class of all ordinals. Let ⟨Kα : α ∈ Ord⟩ be a system of varieties definable by a schema. Given any ordinal α, we define an operator Nrα that acts on Kβ for any β > α giving an algebra in Kα, as an abstraction of taking α-neat reducts for cylindric algebras. We show that for any positive k, and any infinite ordinal α that SNrαKα+k+1 cannot be axiomatized by a finite schema over SNrαKα+k given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of Lω,ω. [ABSTRACT FROM AUTHOR]

The general aim of this article is to study the negation fragment of classical logic within the framework of contemporary (Abstract) Algebraic Logic. More precisely, we shall find the three classes of algebras that are canonically associated with a logic in Algebraic Logic, i.e. we find the classes |$\textrm{Alg}^*$|⁠ , |$\textrm{Alg}$| and the intrinsic variety of the negation fragment of classical logic. In order to achieve this, firstly, we propose a Hilbert-style axiomatization for this fragment. Then, we characterize the reduced matrix models and the full generalized matrix models of this logic. Also, we classify the negation fragment in the Leibniz and Frege hierarchies. [ABSTRACT FROM AUTHOR]

As is well known, any preorder $ R $ R on a set $ U $ U induces an Alexandrov topology on $ U $ U. In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on $ U $ U even if $ R $ R is not a preorder. If this is the case, then we call $ R $ R a crypto-preorder. The paper studies the conditions under which a relation $ R $ R is a crypto-preorder and how to transform a crypto-preorder into a proper preorder. [ABSTRACT FROM AUTHOR]

Residuated lattices are the major algebraic counterpart of logics without contraction rule, as they are more generalized logic systems including important classes of algebras such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and De Morgan residuated lattices among others, on which filters and ideals are sets of provable formulas. This paper presents a meaningful exploration of the topological properties of prime ideals of residuated lattices. Our primary objective is to endow the set of prime ideals with the stable topology, a topological framework that proves to be more refined than the well-known Zariski topology. To achieve this, we introduce and investigate the concept of pure ideals in the general framework of residuated lattices. These pure ideals are intimately connected to the notion of annihilator in residuated lattices, representing precisely the pure elements of quantales. In addition, we establish a relation between pure ideals and pure filters within a residuated lattice, even though these concepts are not dual notions. Furthermore, thanks to the concept of pure ideals, we provide a rigorous description of the open sets within the stable topology. We introduce the i-local residuated lattices along with their properties, demonstrating that they coincide with local residuated lattices. The findings presented in this study represent an extension beyond previous work conducted in the framework of lattices, and classes of residuated lattices. [ABSTRACT FROM AUTHOR]

An algebraic characterisation is given of the Mares-Goldblatt semantics for quantified extensions of relevant and modal logics. Some features of this more general semantic framework are investigated, and the relations to some recent work in algebraic semantics for quantified extensions of non-classical logics are considered. [ABSTRACT FROM AUTHOR]

We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic (CHL ), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for CHL ; moreover, we suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsner's idea of superconnexivity. [ABSTRACT FROM AUTHOR]

We redefine a system of varieties definable by a schema of equations to include finite dimensions. Then we present a technique using ultraproducts enabling one to lift results proved for every finite dimension to the transfinite. Let \(\bf Ord\) denote the class of all ordinals. Let \(\langle \mathbf{K}_{\alpha}: \alpha\in \bf Ord\rangle\) be a system of varieties definable by a schema. Given any ordinal \(\alpha\), we define an operator \(\mathsf{Nr}_{\alpha}\) that acts on \(\mathbf{K}_{\beta}\) for any \(\beta>\alpha\) giving an algebra in \(\mathbf{K}_{\alpha}\), as an abstraction of taking \(\alpha\)-neat reducts for cylindric algebras. We show that for any positive \(k\), and any infinite ordinal \(\alpha\) that \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k+1}\) cannot be axiomatized by a finite schema over \(\mathbf{S}\mathsf{Nr}_{\alpha}\mathbf{K}_{\alpha+k}\) given that the result is valid for all finite dimensions greater than some fixed finite ordinal. We apply our results to cylindric algebras and Halmos quasipolyadic algebras with equality. As an application to our algebraic result we obtain a strong incompleteness theorem (in the sense that validitities are not captured by finitary Hilbert style axiomatizations) for an algebraizable extension of \(L_{\omega,\omega}\).

In this paper we deepen some aspects of the statistical approach to relevance by providing logics for the syntactical treatment of probabilistic relevance relations. Specifically, we define conservative expansions of Classical Logic endowed with a ternary connective → indeed, a constrained material implication -- whose intuitive reading is "x materially implies y and it is relevant to y under the evidence z". In turn, this ensures the definability of a formula in three-variables R(x, z, y) which is the representative of relevance in the object language. We outline the algebraic semantics of such logics, and we apply the acquired machinery to investigate some termdefined weakly connexive implications with some intuitive appeal. As a consequence, a further motivation of (weakly) connexive principles in terms of relevance and background assumptions obtains. [ABSTRACT FROM AUTHOR]

The book "Aristotle's Syllogism and the Creation of Modern Logic: Between Tradition and Innovation, 1820s–1930s" edited by Lukas M. Verburgt and Matteo Cosci is a valuable contribution to the history of logic. The collection of 14 papers aims to reassess the origins of modern formal logic and challenges the common narrative that modern logic began with Gottlob Frege in 1879, erasing lesser-known logicians, including women. The book provides a more accurate and complex narrative of the origins of modern logic by exploring the contributions of various logicians and their relation to traditional syllogistic logic. The chapters are organized chronologically and provide condensed overviews of the logicians' work, allowing for cross-readings and further exploration. However, it should be noted that the book does not discuss Aristotle's logic but focuses on logicians' relation to traditional syllogistic logic in the 19th and 20th centuries. Overall, the book challenges the traditional narrative and offers a richer understanding of the origins of modern logic. [Extracted from the article]

This paper offers a logico-algebraic investigation of AGM belief revision based on the logic of paradox (LP ). First, we define a concrete belief revision operator for LP , proving that it satisfies a generalised version of the traditional AGM postulates. Moreover, we investigate to what extent the Levi and Harper identities, in their classical formulation, can be applied to a paraconsistent account of revision. We show that a generalised Levi-type identity still yields paraconsistent-based revisions that are fully compatible with the AGM postulates. The main outcome is that, once the classical AGM framework is lifted up to an appropriate level of generality, it still appears as a regulative ideal for treating of paraconsistent-based epistemic operators. [ABSTRACT FROM AUTHOR]

In a large number of multivalued logic and fuzzy logic of algebraic systems, residuated lattices play a prominent role and have considerable applications. States operators have been introduced on residuated lattices, and their properties are useful for the development of an algebraic theory of probabilistic models of those algebras. In this paper, we introduce the notion of state ideal in the framework of state residuated lattices, investigate some related properties, and provide several examples. Also, we present two types of state residuated lattices: state i-simple residuated lattices and state i-local residuated lattices, and characterize them. Moreover, the relationship between state ideals and state filters is analyzed using the set of complement elements. Furthermore, we prove that the lattice of all state ideals of a given state residuated lattice is a complete lattice. The notion of obstinate state ideals in state residuated lattices is also introduced, and several characterizations are presented. [ABSTRACT FROM AUTHOR]

This paper focuses on exploring restricted mathematical concepts within the domain of BCK/BCI-algebras, specifically delving into the intricate realm of Multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. BCK and BCI-algebras are pivotal structures in mathematical logic and algebraic systems, finding widespread applications in fields like computer science and artificial intelligence. Our contribution lies in the introduction and thorough investigation of the innovative notions of multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals, uniquely tailored for BCK/BCI-algebras. These ideals exhibit exceptional flexibility in managing uncertain and hesitant information, serving as potent tools for modeling and solving real-world problems characterized by imprecise or incomplete data. This study rigorously defines and explores the foundational properties of multi-polar Q-hesitant fuzzy soft implicative ideals, underscoring their relevance and applicability within BCK/BCI-algebras. Additionally, we present the concept of positive implicative ideals, establishing their interconnectedness with multi-polar Q-hesitant fuzzy soft implicative ideals. Our investigation delves into these ideals' algebraic and logical facets, offering valuable insights into their mutual interactions and significance within the context of BCK/BCI-algebras. To facilitate practical implementation, we develop algorithms and methodologies for identifying and characterizing multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals. These computational tools enable efficient decision-making in scenarios involving uncertainty. Through illustrative examples and case studies, we showcase the potential of these ideals in handling complex, uncertain information, demonstrating their efficacy in aiding problem-solving processes. This research contributes significantly to advancing BCK/BCI-algebra theory by introducing innovative mathematical structures that bridge the gap between fuzzy logic, soft computing, and implicative ideals. The proposed multi-polar Q-hesitant fuzzy soft implicative and positive implicative ideals open new avenues for addressing real-world problems characterized by imprecision and uncertainty. As such, they represent a valuable addition to the field of algebraic structures and their applications. [ABSTRACT FROM AUTHOR]

This document is a special issue of the Journal of Applied Logics, focusing on the frontiers of logic and computation in China. The authors highlight the increasing influence of logic in various fields, including mathematics, physics, computer science, philosophy, cognitive science, and linguistics. Chinese scholars have made significant contributions to logic and computation research, covering topics such as classical and non-classical logic, algebraic logic, temporal logic, probabilistic logic, and knowledge-based systems. The special issue features five contributions that explore different aspects of logic and computation, including the study of algebraic models of modal logic, similarity monadic fuzzy predicate logic, monadic operators on bounded L-algebras, ideals on pseudo equality algebras, and weak hyper filters in hyper BE-algebras. The editors express gratitude to the authors and reviewers for their contributions. [Extracted from the article]

It is shown that in order to increase the efficiency of using methods of abstract algebra in modern information technologies, it is important to establish an explicit connection between operations corresponding to various varieties of multivalued logics and algebraic operations. For multivalued logics, the number of variables in which is equal to a prime number, such a connection is naturally established through explicit algebraic expressions in Galois fields. It is possible to define an algebraic δ-function, which allows you to reduce any truth table to an algebraic expression, for the case when the number of values accepted by a multivalued logic variable is equal to an integer power of a prime number. In this paper, we show that the algebraic δ-function can also be defined for the case when the number of values taken by a multivalued logic variable is p − 1, where p is a prime number. This function also allows to reduce logical operations to algebraic expressions. Specific examples of the constructiveness of the proposed approach are presented, as well as electronic circuits that experimentally prove its adequacy. [ABSTRACT FROM AUTHOR]

The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra. [ABSTRACT FROM AUTHOR]

The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order |$p$| and |$p$| -rings, |$p$| prime and lift this result to the axled Łukasiewicz–Moisil algebra |$L \cong B_s \times P$| and the ring |$\prod ^s F_2 \times \prod ^l F_p$|⁠ , where |$B_s$| is a Boolean algebra of order |$2^s$|⁠ , |$P$| a |$p$| -valued Post algebra of order |$p^l$| and |$F_p$| is the prime field of order |$p$|⁠. [ABSTRACT FROM AUTHOR]

In this paper we introduce the modelwise interpolation property of a logic that states that whenever \(\models\phi\to\psi\) holds for two formulas \(\phi\) and \(\psi\), then for every model \(\mathfrak{M}\) there is an interpolant formula \(\chi\) formulated in the intersection of the vocabularies of \(\phi\) and \(\psi\), such that \(\mathfrak{M}\models\phi\to\chi\) and \(\mathfrak{M}\models\chi\to\psi\), that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic.

In this paper we propose an algebraic model for a modal epistemic logic. Although it is known the existence of algebraic models for modal logics, considering that there are so many different modal logics, so it is not usual to give an algebraic model for each such system. The basic epistemic logic used in the paper is bimodal and we can show that the epistemic algebra introduced in the paper is an adequate model for it.

This text proposes an innovative reconstruction of Aristotle's logical system within the reinterpretative line of Smiley and Corcoran based on algebraic logics, escaping, however, Boethiu's canon by integrating other conceptual tools from the sources, in particular, the use of «negative» terms (here infinite) is proposed together with the concept of «ἐπαγογή» presented in the Posterior Analytics as a heuristical context for the system exposed in the Prior Analytics, in which explicit relationships between two universes of terms representing the intensional and extensional realms are established, generating a framework where the ecthetic proof method can be organically incorporated, resulting from its formalization and computable structuralization a more fitter non-classical logical language for syllogistic inference. [ABSTRACT FROM AUTHOR]

Default Logics are a family of non-monotonic formalisms having so-called defaults and extensions as their common foundation. Traditionally, default logics have been defined and dealt with via syntactic notions of consequence in propositional or first-order logic. Here, we build default logics on modal logics. First, we present these default logics syntactically. Then, we elaborate on an algebraic counterpart. More precisely, we extend the notion of a modal algebra to accommodate for defaults and extensions. Our algebraic view of default logics concludes with an algebraic completeness result and a way of comparing default logics borrowing ideas from the concept of bisimulation in modal logic. To our knowledge, this take on default logics approach is novel. Interestingly, it also lays the groundwork for studying default logics from a dynamic logic perspective. [ABSTRACT FROM AUTHOR]

We propose a new modal logic endowed with a simple deductive system to interpret Aristotle's theory of the modal syllogism. While being inspired by standard propositional modal logic, it is also a logic of terms that admits a (sound) extensional semantics involving possible states-of-affairs in a given world. Applied to the analysis of Aristotle's modal syllogistic as found in the Prior Analytics A8-22, it sheds light on various fine-grained distinctions which when made allow us to clarify some ambiguities and obtain a completely consistent system and prove all of the modal syllogisms considered valid by Aristotle. This logic allows us also to make a connection with the axioms of modern propositional modal logic and to perceive to what extent these are implicit in Aristotle's reasoning. Further work will involve addressing the question of the completeness of this logic (or variants thereof) together with an extension of the logic to include a calculus of relations (for instance the relational syllogistic treated in Galen's Introduction to Logic) which Slomkowsky has argued is already found in the Topics. [ABSTRACT FROM AUTHOR]

It is introduced a new algebra (A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) called L P G -algebra if (A , ⊗ , ⊕ , ∗ , 0 , 1) is L P -algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and (A , ⇀ , 0 , 1) is a Gödel algebra (i.e. Heyting algebra satisfying the identity (x ⇀ y) ∨ (y ⇀ x) = 1) . The lattice of congruences of an L P G -algebra (A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra (A , ⊗ , ⊕ , ∗ , 0 , 1) . The variety L P G of L P G -algebras is generated by the algebras (C , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1) where (C , ⊗ , ⊕ , ∗ , 0 , 1) is Chang MV-algebra. Any L P G -algebra is bi-Heyting algebra. The set of theorems of the logic L P G is recursively enumerable. Moreover, we describe finitely generated free L P G -algebras. [ABSTRACT FROM AUTHOR]

This comparative study gives ancillary elements supporting the leading theories on Babylonian music, showing the relationship between the numerical series and several resulting geometrical figures, which logically precede the writing of the tablet(s) and were probably part of the musician's analysis and reasoning as "implied elements" of the same tablets. Taking into consideration Cicero's Somnium Scipionis describing the orbits' sounds, a few ancillary elements have emerged, highlighting (from a different perspective) innovative - but foreseeable - aspects of the Mesopotamian algebraic logic applying to music, suggesting additional uses of the two texts at issue, in conformity with the leading scholars' opinions. New graphic representations of the tablet are provided. This work should be meant as a preliminary note underlining new data. Further interdisciplinary studies and research (on Babylonian mathematics, archaeology, archaeo-musicology; astronomy) will be necessary in future to confirm, expand or question the results discussed in this work. [ABSTRACT FROM AUTHOR]

We use Constraint Satisfaction Methods to construct and enumerate finite L-algebras up to isomorphism. These objects were recently introduced by Rump and appear in Garside theory, algebraic logic, and the study of the combinatorial Yang–Baxter equation. There are 377,322,225 isomorphism classes of L-algebras of size eight. The database constructed suggests the existence of bijections between certain classes of L-algebras and well-known combinatorial objects. We prove that Bell numbers enumerate isomorphism classes of finite linear L-algebras. We also prove that finite regular L-algebras are in bijective correspondence with infinite-dimensional Young diagrams. [ABSTRACT FROM AUTHOR]

Let \(2

In this paper we introduce the modelwise interpolation property of a logic that states that whenever |= ϕ → ψ holds for two formulas ϕ and ψ, then for every model M there is an interpolant formula χ formulated in the intersection of the vocabularies of ϕ and ψ, such that M |= ϕ → χ and M |= χ → ψ, that is, the interpolant formula in Craig interpolation may vary from model to model. We compare the modelwise interpolation property with the standard Craig interpolation and with the local interpolation property by discussing examples, most notably the finite variable fragments of first order logic, and difference logic. As an application we connect the modelwise interpolation property with the local Beth definability, and we prove that the modelwise interpolation property of an algebraizable logic can be characterized by a weak form of the superamalgamation property of the class of algebras corresponding to the models of the logic. [ABSTRACT FROM AUTHOR]

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. [ABSTRACT FROM AUTHOR]

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In the Laws of Thought, Boole establishes a theory of secondary propositions based upon the notion of time. This temporal interpretation of secondary propositions has historically been met with wide disapproval and is usually dismissed in the modern literature as a philosophical non-starter. What was Boole thinking? This paper attempts to give an answer to this question. Specifically, it provides an account according to which Boole's temporal interpretation follows from his psychologistic conception of logic, in addition to certain background assumptions regarding the psychological necessity of the concept of time. Once Boole's psychological premises are laid bare, it becomes clearer how he might have viewed the temporal interpretation to be an essential feature of his theory of secondary propositions. [ABSTRACT FROM AUTHOR]

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality κ , where κ is a regular cardinal. The corresponding new notion is called κ -filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different κ -filter pairs give rise to a fixed logic of cardinality κ . To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality κ . Along the way we use κ -filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair. [ABSTRACT FROM AUTHOR]

The Boolean models of argumentation semantics have been established in various ways. These models commonly translate the conditions of extension-based semantics into some constraints of the models. The goal of this work is to explore a simple method to build Boolean models for argumentation. In this paper, the attack relation is treated as an operator, and its value is calculated by the values of its target and source arguments. By examining the values of the attacks, a Boolean model of conflict-free sets is introduced. This novel method simplifies the existing ways by eliminating the various constraints. The conflict-free sets can be calculated by simply checking the values of the attacks. [ABSTRACT FROM AUTHOR]

The category LAlg of L-algebras is shown to be complete and cocomplete, regular with a zero object and a projective generator, normal and subtractive, ideal determined, but not Barr-exact. Originating from algebraic logic, L-algebras arise in the theory of Garside groups, measure theory, functional analysis, and operator theory. It is shown that the category LAlg is far from protomodular, but it has natural semidirect products which have not been described in category-theoretic terms. [ABSTRACT FROM AUTHOR]

The article discusses the use of formalization and mathematical modeling in macroeconomics, particularly in relation to the work of John Maynard Keynes. It highlights the debate surrounding the use of economic analytics versus verbal and logical exposition. The author, Peter Skott, argues for the need for formalization in modern macroeconomics but criticizes the logical flaws and lack of empirical evidence in current models. Skott advocates for a structural and behavioral approach that incorporates institutional changes and findings from behavioral economics. The book provides detailed analytical treatments of specific issues and challenges macro theorists and modelers to re-examine their assumptions. [Extracted from the article]

P. Hájek introduced an S5-like modal fuzzy logic S5(BL) and showed that is equivalent to the monadic basic predicate logic mBL∀. Inspired by the above important results, D. Castaño et al. introduced monadic BL-algebras and their corresponding propositional logic S5'(BL), which is a simplified set of axioms of S5(BL). In this paper, we review the algebraic semantics of S5'(BL) and obtain some new results regarding to monadic BL-algebras. First we recall that S5'(BL) is completeness with respect to the variety 필픹핃 of monadic BL-algebras and obtain a necessary and sufficient condition for the logic S5'(BL) to be semilinear. Then we study some further algebraic properties of monadic BL-algebras and discuss the relationship between monadic MV-algebras and monadic BL-algebras. Finally we give some characterizations of representable, simple, semisimple and directly indecomposable monadic BL-algebras, which are important members of the variety 필픹핃. These results also constitute a crucial first step for providing an equivalent algebraic foundation for mBL∀. [ABSTRACT FROM AUTHOR]

In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called |$R$| -models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups. [ABSTRACT FROM AUTHOR]