We find the model completion of the theory modules over ������, where ������ is a finitely generated commutative algebra over a field K. This is done in a context where the field K and the module are represented by sorts in the theory, so that constructible sets associated with a module can be interpreted in this language. The language is expanded by additional sorts for the Grassmanians of all powers of $K^n $ , which are necessary to achieve (...) quantifier elimination. The result turns out to be that the model completion is the theory of a certain class of "big" injective modules. In particular, it is shown that the class of injective modules is itself elementary. We also obtain an explicit description of the types in this theory. (shrink)

This main aim of this paper is to prove that in a direct commutative BCK-algebra an ideal I is nitely generated if and only if I is a principal ideal. This result generalizes the result obtained by E. Y. Deeba in [2]. We also give an answer to the question posed by E. Y. Deeba in [1]: for what class of BCK-algebras is every Noetherian algebra a principal ideal algebra ?

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. Łukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several (...) recent papers and includes the class of MV-algebras. We show that the logic L CBA is very close to the Łukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization. (shrink)

We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization (...) of approximability of a locally compact system A by systems from K using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula φ holds in A if and only if all precise enough approximations of φ hold in all precise enough approximations of A. We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings. (shrink)

In this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S'(ai, l) for $l < n$. Then we derive two important theorems. One (...) claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ∈ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings. (shrink)

In this paper we investigate the class of MV-algebras equipped with two quantifiers which commute as a natural generalization of diagonal-free two-dimensional cylindric algebras. In the 40s, Tarski first introduced cylindric algebras in order to provide an algebraic apparatus for the study of classical predicate calculus. The diagonal–free two-dimensional cylindric algebras are special cylindric algebras. The treatment here of MV-algebras is done in terms of implication and negation. This allows us to simplify some results due to Di Nola and Grigolia (...) :125–139, 2004) related to the characterization of a quantifier in terms of some special sub-algebra associated to it. On the other hand, we present a topological duality for this class of algebras and we apply it to characterize the congruences of one algebra via certain closed sets. Finally, we study the subvariety of this class generated by a chain of length n + 1. We prove that the subvariety is semisimple and we characterize their simple algebras. Using a special functional algebra, we determine all the simple finite algebras of this subvariety. (shrink)

In the paper it will be shown that Reichenbach’s Weak Common Cause Principle is not valid in algebraic quantum field theory with locally finite degrees of freedom in general. Namely, for any pair of projections A, B supported in spacelike separated double cones ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ , respectively, a correlating state can be given for which there is no nontrivial common cause (system) located in the union of the backward light cones of ${\mathcal{O}}_{a}$ and ${\mathcal{O}}_{b}$ and commuting with the (...) both A and B. Since noncommuting common cause solutions are presented in these states the abandonment of commutativity can modulate this result: noncommutative Common Cause Principles might survive in these models. (shrink)

In the present contribution we look at the legacy of Hilbert's programme in some recent developments in mathematics. Hilbert's ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been a source of motivation for the so--called reverse mathematics programme initiated by H. Friedman and S. Simpson. More recently Hilbert's programme has inspired T. Coquand and H. Lombardi to undertake a new approach to constructive algebra in which strong emphasis is laid (...) on the use of finite methods. The main aim is to eliminate the ideal objects and in so doing obtain more elementary and informative proofs. We survey some work in commutative algebra---mainly about and around the Zariski spectrum and the Krull dimension of a commutative ring---which witnesses the feasibility of such a revised Hilbert's programme. (shrink)

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some (...) richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (shrink)

We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17—19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FL i -algebras and FL o -algebras. On the other hand, we show that the subvariety lattice of residuated lattices contains only five 3-potent commutative representable atoms (...) and two integral commutative representable atoms. Inspired by the construction of atoms, we are also able to prove that the variety of integral commutative representable residuated lattices is generated by its 1-generated finite members. (shrink)

A systematic approach to the algebraic and Kripke semantics for logics with restricted structural rules, notably for logics on an underlying non-distributive lattice, is developed. We provide a new topological representation theorem for general lattices, using the filter space X. Our representation involves a galois connection on subsets of X, hence a closure operator $\Gamma$, and the image of the representation map is characterized as the collection of $\Gamma$-stable, compact-open subsets of the filter space . The original lattice ${\cal L}$ (...) is thus imbedded in the complete lattice of stable sets. The representation is shown to be functorial and is extended to a Stone-type duality. ;In the presence of additional operators, the extension problem is addressed in terms of what we call adjoint completions of normal operators. An operator f on a lattice L is normal if f: $L\sp{} \times \cdots \times L\sp{} \to L\sp{}$ is a monotone map, where each $L\sp{}$, $i \in n$ + 1, is either L or $L\sp{op}$ and f sends finite joins of $L\sp{}$ to joins in $L\sp{}$. Every familiar operator, such as $\circ$ , + , $\gets, \to, \neg, \square$ O is normal. By a uniform argument we show that normal operators $f\sb{i}$ on L extend to operators $F\sb{i}$ on $\Gamma X$ that distribute over arbitrary joins or meets and that \sb{i\in I})$ is a universal such extension. ;We also raise and resolve the question of representability of operators on stable sets by relations. We give an extensive number of examples of representation of specific operators and provide explicit constructions of canonical Kripke frames, based on our extension arguments, for a number of logical systems, notably for Intuitionistic and Classical Linear Logics, or Non-commutative Linear Logic and of fragments thereof. (shrink)

In this paper we characterize the MV-algebras containing as subalgebras Post algebras of finitely many orders. For this we study cyclic elements in MV-algebras which are the generators of the fundamental chain of the Post algebras.

Pseudo MV-algebras are a non-commutative extension of MV-algebras introduced recently by Georgescu and Iorgulescu. We introduce states (finitely additive probability measures) on pseudo MV-algebras. We show that extremal states correspond to normal maximal ideals. We give an example in that, in contrast to classical MV-algebras introduced by Chang, states can fail on pseudo MV-algebras. We prove that representable and normal-valued pseudo MV-algebras admit at least one state.

A pO -algebra $${(L; f, \, ^{\star})}$$ is an algebra in which ( L ; f ) is an Ockham algebra, $${(L; \, ^{\star})}$$ is a p -algebra, and the unary operations f and $${^{\star}}$$ commute. Here we consider the endomorphism monoid of such an algebra. If $${(L; f, \, ^{\star})}$$ is a subdirectly irreducible pK 1,1 - algebra then every endomorphism $${\vartheta}$$ is a monomorphism or $${\vartheta^3 = \vartheta}$$ . When L is (...) class='Hi'>finite the endomorphism monoid of L is regular, and we determine precisely when it is a Clifford monoid. (shrink)

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional (...) non-associative algebras over an algebraically closed field of characteristic [Formula: see text] give rise to triangular rings without the CBP. This also applies to Baudisch’s [Formula: see text]-step nilpotent Lie algebras, which yields the existence of a [Formula: see text]-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP. (shrink)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras, where * is a continuous t-norm. In this paper we investigate the structure (...) of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that ([ 0,1], *, 1) is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and ([ 0,1], *, →, 1) becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras ([ 0,1], *, →, (...) 1), where * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some (...) richer subreducts of the class of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Ekstein has shown that causal independence neither implies nor is implied by commutativity in an infinite-dimensional, reducible construction. DeFacio and Taylor have presented a finite-dimensional irreducible example of Ekstein's proposition. Avishai and Ekstein have shown that the original question regarding locality for algebraic quantum field theories remainsopen. We concur with that claim and offer additional arguments. A new denumerably infinite-dimensional, irreducible example is presented here which shows that a sort of “orthogonality” among operators is involved. Some observations on localC*-andW*-algebras (...) are given. (shrink)

In this paper we study a certain class of Lie algebras over commutative von Neumann algebras satisfying a certain finiteness condition. By using Boolean valued methods developed by Takeuti [8]-[11], we will establish the basic structure and representation theorems.

We consider countable so‐called rich subsemigroups of ; each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, if and only if is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic (...) equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented. (shrink)

The IKt-algebras were introduced in the paper An algebraic axiomatization of the Ewald’s intuitionistic tense logic by the first and third author. In this paper, our main interest is to investigate the principal and Boolean congruences on IKt-algebras. In order to do this we take into account a topological duality for these algebras obtained in Figallo et al. :673–701, 2017). Furthermore, we characterize Boolean and principal IKt-congruences and we show that Boolean IKt-congruence are principal IKt-congruences. Also, bearing in mind the (...) above results, we obtain that Boolean IKt-congruences are commutative, regular and uniform. Finally, we characterize the principal IKt-congruences in the case that the IKt-algebra is linear and complete whose prime filters are complete and also the case that it is linear and finite. This allowed us to establish that the intersection of two principal IKt-congruences on these algebras is a principal one and also to determine necessary and sufficient conditions so that a principal IKt-congruence is a Boolean one on theses algebras. (shrink)

Fix a finite ordinal \ and let \ be an arbitrary ordinal. Let \ denote the class of cylindric algebras of dimension \ and \ denote the class of relation algebras. Let \\) stand for the class of polyadic algebras of dimension \. We reprove that the class \ of completely representable \s, and the class \ of completely representable \s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \ between polyadic (...) algebras of dimension \ and diagonal free \s. We show that that the class of completely and strongly representable algebras in \ is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \ is not closed under \. In contrast, we show that given \, and an atomic \, then for any \, \ is a completely representable \. We show that for any \, the class of completely representable algebras in certain reducts of \s, that happen to be varieties, is elementary. We show that for \, the the class of polyadic-cylindric algebras dimension \, introduced by Ferenczi, the completely representable algebras coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \ and \s for \ an infinite ordinal, proving negative results for the first and positive ones for the second. (shrink)

In this paper, we define the notion of PC-lattice, as a generalization of finite positive implicative BCK-algebras with condition and bounded commutative BCK-algebras. We investiate some results for Pc-lattices being a new class of BCK-lattices. Specially, we prove that any Boolean lattice is a PC-lattice and we show that if X is a PC-lattice with condition S, then X is an involutory BCK-algebra if and only if X is a commutative BCK-algebra. Finally, we prove that (...) any PC-lattice with condition is a distributive BCK-algebra. (shrink)

The notion of monadic three-valued ukasiewicz algebras was introduced by L. Monteiro ([12], [14]) as a generalization of monadic Boolean algebras. A. Monteiro ([9], [10]) and later L. Monteiro and L. Gonzalez Coppola [17] obtained a method for the construction of a three-valued ukasiewicz algebra from a monadic Boolea algebra. In this note we give the construction of a monadic three-valued ukasiewicz algebra from a Boolean algebra B where we have defined two quantification operations and * (...) such that *x=*x (where *x=-*-x). In this case we shall say that and * commutes. If B is finite and is an existential quantifier over B, we shall show how to obtain all the existential quantifiers * which commute with .Taking into account R. Mayet [3] we also construct a monadic three-valued ukasiewicz algebra from a monadic Boolean algebra B and a monadic ideal I of B. (shrink)

For an arbitrary finite algebra $\g A = (A, f, 0, 1)$ one defines a double sequence $a(i,j)$ by $a(i,0)\!=\!a(0,j)\! =\! 1$ and $a(i,j) \!= \!f( a(i, j-1) , a(i-1,j) )$.The problem if such recurrent double sequences are ultimately zero is undecidable, even if we restrict it to the class of commutative finite algebras.

We will show that almost all nonassociative relation algebras are symmetric and integral (in the sense that the fraction of both labelled and unlabelled structures that are symmetric and integral tends to $1$ ), and using a Fraïssé limit, we will establish that the classes of all atom structures of nonassociative relation algebras and relation algebras both have $0$ – $1$ laws. As a consequence, we obtain improved asymptotic formulas for the numbers of these structures and broaden some known probabilistic (...) results on relation algebras. (shrink)

Simple observations on diophantine definability over finite commutative rings lead to a characterization of those rings in terms of their diophantine behavior.

One kind of generalized measures called quantum measures on finite effect algebras, which fulfil the grade-2 additive sum rule, is considered. One basis of vector space of quantum measures on a finite effect algebra with the Riesz decomposition property (RDP for short) is given. It is proved that any diagonally positive symmetric signed measure \(\lambda \) on the tensor product \(E\otimes E\) can determine a quantum measure \(\mu \) on a finite effect algebra \(E\) with (...) the RDP such that \(\mu (x)=\lambda (x\otimes x)\) for any \(x\in E\) . Furthermore, some conditions for a grade-2 additive measure \(\mu \) on a finite effect algebra \(E\) are provided to guarantee that there exists a unique diagonally positive symmetric signed measure \(\lambda \) on \(E\otimes E\) such that \(\mu (x)=\lambda (x\otimes x)\) for any \(x\in E\) . At last, it is showed that any grade- \(t\) quantum measure on a finite effect algebra \(E\) with the RDP is essentially established by the values on a subset of \(E\). (shrink)

Fix 2 < n < ω. Let CA n denote the class of cylindric algebras of dimension n, and let RCA n denote the variety of representable CA n ’s. Let L n denote first-order logic restricted to the first n variables. Roughly, CA n, an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of L n, namely, its proof theory, while RCA n algebraically and geometrically represents the Tarskian semantics of L n. Unlike Boolean (...) algebras having a Stone representation theorem, RCA n ⊊ CA n. Using combinatorial game theory, we show that the existence of certain finite relation algebras RA, which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for L n. Using special cases of such finite RA ’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on RCA n and we re-prove Hirsch and Hodkinson’s result that the class of completely representable CA n ’s is not first-order definable. We show that if T is an L n countable theory that admits elimination of quantifiers, λ is a cardinal < 2 ℵ 0, and F = 〈 Γ i : i < λ 〉 is a family of complete nonprincipal types, then F can be omitted in an ordinary countable model of T. (shrink)

We define a finite relation algebra and show that the network satisfaction problem is undecidable for this algebra.

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic (...) localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection. (shrink)

The aim of this paper is to characterize varieties of Heyting algebras with decidable theory of their finite members. Actually we prove that such varieties are exactly the varieties generated by linearly ordered algebras. It contrasts to the result of Burris [2] saying that in the case of whole varieties, only trivial variety and the variety of Boolean algebras have decidable first order theories.

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with a decidable theory of modules. We prove that if R is wild, then the theory of all R-modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

The recently developed technique of Bohrification associates to a (unital) C*-algebra Athe Kripke model, a presheaf topos, of its classical contexts;in this Kripke model a commutative C*-algebra, called the Bohrification of A;the spectrum of the Bohrification as a locale internal in the Kripke model. We propose this locale, the ‘state space’, as a (n intuitionistic) logic of the physical system whose observable algebra is A.We compute a site which externally captures this locale and find that externally (...) its points may be identified with partial measurement outcomes. This prompts us to compare Scott-continuity on the poset of contexts and continuity with respect to the C*-algebra as two ways to mathematically identify measurement outcomes with the same physical interpretation. Finally, we consider the not-not-sheafification of the Kripke model on classical contexts and obtain a space of measurement outcomes which for commutative C*-algebras coincides with the spectrum. The construction is functorial on the category of C*-algebras with commutativity reflecting maps. (shrink)

Using the description of the Ziegler spectrum we characterise modules with various stability-theoretic properties (ω-stability, superstability, categoricity) over certain classes of finite-dimensional algebras. We also show that, for modules over the algebras we consider, having few types is equivalent to being ω-stable.

We introduce a family of tableau calculi for deontic action logics based on finite boolean algebras, these logics provide deontic operators which are applied to a finite number of actions ; furthermore, in these formalisms, actions can be combined by means of boolean operators, this provides an expressive algebra of actions. We define a tableau calculus for the basic logic and then we extend this calculus to cope with extant variations of this formalism; we prove the soundness (...) and completeness of these proof systems. In addition, we investigate the computational complexity of the satisfiability problem for DAL and its extensions; we show this problem is NP-complete when the number of actions considered is fixed, and it is \-Hard when the number of actions is taken as an extra parameter. The tableau systems introduced here can be implemented in PSPACE, this seems reasonable taking into consideration the computational complexity of the logics. (shrink)

In this paper we investigate the computational complexity of deciding if the variety generated by a given finite idempotent algebra satisfies a special type of Maltsev condition that can be specified using a certain kind of finite labelled path. This class of Maltsev conditions includes several well known conditions, such as congruence permutability and having a sequence of n Jónsson terms, for some given n. We show that for such “path defined” Maltsev conditions, the decision problem is (...) polynomial-time solvable. (shrink)

Quantum B-algebras, the partially ordered implicational algebras arising as subreducts of quantales, are introduced axiomatically. It is shown that they provide a unified semantic for non-commutative algebraic logic. Specifically, they cover the vast majority of implicational algebras like BCK-algebras, residuated lattices, partially ordered groups, BL- and MV-algebras, effect algebras, and their non-commutative extensions. The opposite of the category of quantum B-algebras is shown to be equivalent to the category of logical quantales, in the way that every quantum B- (...) class='Hi'>algebra admits a natural embedding into a logical quantale, the enveloping quantale. Partially defined products of algebras related to effect algebras are handled efficiently in this way. The unit group of the enveloping quantale of a quantum B-algebra X is shown to be always contained in X, which gives a functorial subgroup X× of X. Similar subfunctors are obtained for the non-commutative extensions of BCK-algebras and effect algebras. The results of Galatos, Jónsson, and Tsinakis on the splitting of generalized BL-algebras into a semidirect product of a partially ordered group operating on an integral residuated poset are extended to a characterization of twisted semidirect products of a po-group by a quantum B-algebra. (shrink)

In this work we introduce a class of commutative rings whose defining condition is that its lattice of ideals, augmented with the ideal product, the semi-ring of ideals, is isomorphic to an MV-algebra. This class of rings coincides with the class of commutative rings which are direct sums of local Artinian chain rings with unit.

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics. The ∨-free reducts of Heyting algebras give rise to (...) the -canonical formulas that we studied in an earlier work. Here we introduce the -canonical formulas, which are obtained from the study of the →-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by -canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas. One of the main ingredients of these formulas is a designated subset D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D=A2, we show that the -canonical formula of A is equivalent to the Jankov formula of A. On the other hand, when D=∅, the -canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics. (shrink)