The theory of boolean algebras with an additional binary operation

This paper deals with Boolean algebras supplied with an additional binary operation, calledB-algebras for short.The aim of the paper is to generalize some theorems concerning topological Boolean algebras to more comprehensive classes ofB-algebras, to formulate fundamental properties ofB-algebras, and to find more important relationships of these algebras to other known algebras.The paper consists of two parts. At the beginning of the first one, several subclasses ofB-algebras are distinguished, and then, their basic properties, connections between them as well as certain relationships with other algebras, are investigated. In particular, it is shown that the class of Boolean algebras together with an arbitrary unary operation is polynomially equivalent to the class ofB ₁-algebras.The second part of the paper is concerned with the theory of filters and congruences inB-algebras.     

B-varieties with normal free algebras

The starting point for the investigation in this paper is the following McKinsey-Tarski's Theorem: if f and g are algebraic functions (of the same number of variables) in a topological Boolean algebra (TBA) and if C(f)C(g) vanishes identically, then either f or g vanishes identically. The present paper generalizes this theorem to B-algebras and shows that validity of that theorem in a variety of B-algebras (B-variety) generated by SCI _B -equations implies that its free Lindenbaum-Tarski's algebra is normal. This is important in the semantical analysis of SCI _B (the Boolean strengthening of the sentential calculus with identity, SCI) since normal B-algebras are just models of this logic. The rest part of the paper is concerned with relationships between some closure systems of filters, SCI _B -theories, B-varieties and closed sets of SCI _B -equations that have been derived both from the semantics of SCI _B and from the semantics of the usual equational logic.To the memory of Jerzy Supecki     

W-algebras which are Boolean products of members of SR[1] and CW-algebras

We show that the class of all isomorphic images of Boolean Products of members of SR [1] is the class of all archimedean W-algebras. We obtain this result from the characterization of W-algebras which are isomorphic images of Boolean Products of CW-algebras.     

Averaging the truth-value in Łukasiewicz logic

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.     

The countable homogeneous universal model of B 2

We give a detailed account of the Algebraically Closed and Existentially Closed members of the second Lee class B ₂ of distributive p-algebras, culminating in an explicit construction of the countable homogeneous universal model of B ₂. The axioms of Schmid [7], [8] for the AC and EC members of B ₂ are reduced to what we prove to be an irredundant set of axioms. The central tools used in this study are the strong duality of Clark and Davey [3] for B ₂ and the method of Clark [2] for constructing AC and EC algebras using a strong duality. Applied to B ₂, this method transfers the entire discussion into an equivalent dual category X ₂ of Boolean spaces which carry a pair of tightly interacting orderings. The doubly ordered spaces of X ₂ prove to be much more readily constructed and analyzed than the corresponding algebras in B ₂.     

Pretabular varieties of modal algebras

We study modal logics in the setting of varieties of modal algebras. Any variety of modal algebras generated by a finite algebra — such, a variety is called tabular — has only finitely many subvarieties, i.e. is of finite height. The converse does not hold in general. It is shown that the converse does hold in the lattice of varieties of K4-algebras. Hence the lower part of this lattice consists of tabular varieties only. We proceed to show that there is a continuum of pretabular varieties of K4-algebras — those are the non-tabular varieties all of whose proper subvarieties are tabular — in contrast with Maksimova's result that there are only five pretabular varieties of S4-algebras.     

N-Valued Logics and Łukasiewicz–Moisil Algebras

Fundamental properties of N-valued logics are compared and eleven theorems are presented for their Logic Algebras, including ?ukasiewicz–Moisil Logic Algebras represented in terms of categories and functors. For example, the Fundamental Logic Adjunction Theorem allows one to transfer certain universal, or global, properties of the Category of Boolean Algebras, , (which are well-understood) to the more general category ${\cal L}$ M _n of ?ukasiewicz–Moisil Algebras. Furthermore, the relationships of LM _n -algebras to other many-valued logical structures, such as the n-valued Post, MV and Heyting logic algebras, are investigated and several pertinent theorems are derived. Applications of ?ukasiewicz–Moisil Algebras to biological problems, such as nonlinear dynamics of genetic networks – that were previously reported – are also briefly noted here, and finally, probabilities are precisely defined over LM _n -algebras with an eye to immediate, possible applications in biostatistics.     

Semisimplicity,EDPC and Discriminator Varieties of Residuated Lattices

We prove that all semisimple varieties of FL _ew-algebras are discriminator varieties. A characterisation of discriminator and EDPC varieties of FL _ew-algebras follows. It matches exactly a natural classification of logics over FL _ew proposed by H. Ono.     

Simplified semantics for relevant logics (and some of their rivals)

This paper continues the work of Priest and Sylvan inSimplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such asB ⁺ andB. We show that the simplified semantics can also be used for a large number of extensions of the positive base logicB ⁺, and then add the dualising^* operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.     

Representations of monadic <Emphasis Type="Italic">MV</Emphasis> -algebras

Representations of monadic MV -algebra, the characterization of locally finite monadic MV -algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV -algebras are given. Moreover, it is shown that finitely generated m-relatively complete subalgebra of finitely generated free MV -algebra is projective.     

Assessment of tensile interphase profile of PVC/TiO2 polymer nanocomposites

The tensile properties of polymer nanocomposites depend dominantly on their interphase profile. The interfacial layer adhesion B-parameter, the relative interphase tensile strength (σ_IR) and the Z-interphase parameter for the central layer are quantified for assessment of tensile interphase properties of PVC/TiO₂ polymer nanocomposites. The redeveloped Pukanszky model is employed to determine the layer adhesion B-parameter of interphases. The study shows that the tensile strength of the interphase σ_I, as well as σ_IR and the B- and Z-parameters of the central layer of the interphase, are significantly correlated with the tensile strength of the nanocomposites.     

Interfacial precipitation of the M5B3-type boride in Ni-based superalloys

Microstructural characteristics of an experimental Ni-based superalloy with boron addition subjected to a long-term ageing treatment were systematically investigated by various kinds of transmission electron microscopy technique. Based on detailed electron diffraction analyses, we found that there are many nanosized M₅B₃ precipitates in our long-term ageing alloys, which keeps a good orientation relationship with the γ/γ′ matrix. Furthermore, the precipitation characteristics of M₅B₃ phase were clarified. It is found that the M₅B₃-type boride prefers to precipitate at the γ/γ′ interfaces and low-angle grain boundaries. These interfacial nanosized precipitates can play the role of pinning effect and are expected to be advantageous for postponing the γ′ rafting and low-angle grain boundary migration to some extent at high temperature.     

On Axiomatizing Shramko-Wansing’s Logic

This work treats the problem of axiomatizing the truth and falsity consequence relations, ⊨ _t and ⊨ _f , determined via truth and falsity orderings on the trilattice SIXTEEN ₃ (Shramko and Wansing, 2005). The approach is based on a representation of SIXTEEN ₃ as a twist-structure over the two-element Boolean algebra.     

Constructive Logic with Strong Negation is a Substructural Logic. II

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL _ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. Presented by Heinrich Wansing     

A Nelsonian Response to ‘the Most Embarrassing of All Twelfth-century Arguments’

Alberic of Paris put forward an argument, ‘the most embarrassing of all twelfth-century arguments’ according to Christopher Martin, which shows that the connexive principles contradict some other logical principles that have become deeply entrenched in our most widely accepted logical theories. Building upon some of Everett Nelson’s ideas, we will show that the steps in Alberic of Paris’ argument that should be rejected are precisely the ones that presuppose the validity of schemas that are nowadays taken as some of the most trivial logical truths: (A∧B) →_AB A and (A∧B) →_AB B, i.e. Simplification.     

On the logic of conscious belief

In this paper we are discussing a version of propositional belief logic, denoted by LB, in which so-called axioms of introspection (B BB and B B B) are added to the usual ones. LB is proved to be sound and complete with respect to Boolean algebras equipped with proper filters (Theorem 5). Interpretations in classical theories (Theorem 4) are also considered. A few modifications of LB are further dealt with, one of which turns out to be S5.     

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

Merging Inference and Conjecture by Information

The intuitive notion of a binary relation on information-bearers, comparingthem with respect to their closeness to the available information, is oftenconstrued in terms of comparing their symmetric difference with, orcompositional similarity to, the available information. This happens forinstance in some treatments of verisimilitude. We expound an abstractmathematical rendering of the relevant data-dependent relation in theframework of Boolean algebras. For every element t of a Boolean algebra B we construct the t-modulated Boolean algebra B_tin which the order relation represents `is at most as compatible with t as' or `is at best as similar to t as'. In the case of Lindenbaum-Tarskialgebras, t expresses the available information, and the compatibilityrelation turns out to be an entwinement of inferential and conjecturalrelations. It is just classical entailment when no information is available(i.e., when t is logically true) and becomes more boldly abductive themore information is available. The rich algebraic structures of a Boolean algebra –- including its Boolean group structures –- play a significant role in this combination of deduction and abduction and also induce cautious anddaring variants of the compatibility relation. Links with the literature onverisimilitude, abduction, and related topics are indicated.     

Maximal Subalgebras of MVn-algebras. A Proof of a Conjecture of A. Monteiro

For each integer n ≥ 2, MV_n denotes the variety of MV-algebras generated by the MV-chain with n elements. Algebras in MV_n are represented as continuous functions from a Boolean space into a n-element chain equipped with the discrete topology. Using these representations, maximal subalgebras of algebras in MV_n are characterized, and it is shown that proper subalgebras are intersection of maximal subalgebras. When A ∈ MV₃, the mentioned characterization of maximal subalgebras of A can be given in terms of prime filters of the underlying lattice of A, in the form that was conjectured by A. Monteiro. Mathematics Subject Classification (2000): 06D30, 06D35, 03G20, 03B50, 08A30. Presented by Daniele Mundici     

Structural stability of W2B5 under high pressure

High-pressure structural stability studies have been carried out on tungsten boride W₂B₅ up to maximum pressure of 36 GPa using a Mao-Bell diamond-anvil cell at beamline BR-12 of the ELETTRA synchrotron facility (λ = 0.68881 Å). The hexagonal phase (S.G:P6₃/mmc) of W₂B₅ is stable up to the maximum pressure studied. The bulk modulus is estimated to be ~347 GPa using the Birch–Murnaghan equation of state. The variation of lattice parameters and bond lengths B–B and W–B have been studied and the c-axis is seen to be marginally more compressible than the a-axis.