Fuzzy Boolean algebras

The purpose of this paper is to generalize the following situation: from the concrete structure $\text{B}$, we define the notion of Boolean algebras; the Stone representation theorem allows us to replace the algebraic study of Boolean algebras by a topological one. Let E be a non-empty set, and J a non-empty ordered set. Note $\text{B}$ the set of all fuzzy subsets of (E,J). We shall introduce the concept of fuzzy Boolean algebra and find a representation theorem. But it will be difficult to speak of the dual fuzzy topological space of a fuzzy Boolean algebra as we shall see further, except in certain particular cases.     

On completeness of locally-internal categories

Locally-internal categories over a topos E are regarded as categories enriched in the bicategory Span E. In this paper we develop some aspects of completeness. For locally-internal categories, completeness means: finite completeness over each fiber, transition functors along the maps of the base topōs, and a Beck-Chevalley condition (see Bénabou, C.R. Acad. Sci. Paris 281 (1975) A897–900). We prove that this notion can be obtained by particularizing to Span E the general notion of completeness of enriched category theory, given in terms of indexed limits. We give also an adjoint functor theorem.     

Equational type characterization for σ-complete MV-algebras

In the framework of algebras with infinitary operations, an equational base for the category of σ-complete MV-algebras is given. In this way, we study some particular objects as simple algebras, directly irreducible algebras, injectives, etc. A completeness theorem with respect to the standard MV-algebra, considered as σ-complete MV-algebra, is obtained. Finally, we apply this result to the study of σ-complete Boolean algebras and σ-complete product MV-algebras.     

An Alternative Definition of Quantifiers on Four-Valued Łukasiewicz Algebras

An alternative notion of an existential quantifier on four-valued ?ukasiewicz algebras is introduced. The class of four-valued ?ukasiewicz algebras endowed with this existential quantifier determines a variety which is denoted by \(\mathbb {M}_{\frac{2}{3}}\mathbb {L}_4\). It is shown that the alternative existential quantifier is interdefinable with the standard existential quantifier on a four-valued ?ukasiewicz algebra. Some connections between the new existential quantifier and the existential quantifiers defined on bounded distributive lattices and Boolean algebras are given. Finally, a completeness theorem for the monadic four-valued ?ukasiewicz predicate calculus corresponding to the dual of the alternative existential quantifier is proven.     

Boolean-like algebras

Using Vaggione’s concept of central element in a double-pointed algebra, we introduce the notion of Boolean-like variety as a generalisation of Boolean algebras to an arbitrary similarity type. Appropriately relaxing the requirement that every element be central in any member of the variety, we obtain the more general class of semi-Boolean-like varieties, which still retain many of the pleasing properties of Boolean algebras. We prove that a double-pointed variety is discriminator if and only if it is semi-Boolean-like, idempotent, and 0-regular. This theorem yields a new Maltsev-style characterisation of double-pointed discriminator varieties.     

Banaschewski’s theorem for S-posets: regular injectivity and completeness

In this paper we study the notion of injectivity in the category Pos-S of S-posets for a pomonoid S. First we see that, although there is no non-trivial injective S-poset with respect to monomorphisms, Pos-S has enough (regular) injectives with respect to regular monomorphisms (sub S-posets). Then, recalling Banaschewski’s theorem which states that regular injectivity of posets with respect to order-embeddings and completeness are equivalent, we study regular injectivity for S-posets and get some homological classification of pomonoids and pogroups. Among other things, we also see that regular injective S-posets are exactly the retracts of cofree S-posets over complete posets.     

Completezza degli spaziL p per misure finitamente additive

Summary A criterion of completeness of metric (not necessarily normed) L^p spaces with respect to bounded finitely additive measures is established (L ^p spaces are defined differently that it is usually done in the setting of finitely additive measures). This result hinges on the Stone representation theorem for Boolean algebras.     

Diads and their Application to Topoi

It is well known that the category of coalgebras for a finite-limit preserving comonad on a topos is again a topos, and the category of algebras for a finite-limit preserving monad is a topos if the monad is idempotent, but not in general. A generalisation of this result (Paré et al., Bull Aus Math Soc 39(3):421–431, 1989) is that the full subcategory of fixed points for any idempotent finite-limit preserving endofunctor is again a topos (and indeed a subquotient in the category of topoi and geometric morphisms). Here, we present a common generalisation of all the above results, based on a notion which we call a diad, which is a common generalisation of a monad and a comonad. Many of the constructions that can be applied to monads and comonads can be extended to all diads. In particular, the category of algebras or coalgebras can be generalised to a category of dialgebras for a diad. The generalisation we present here is that the category of dialgebras for a finite-limit preserving left diad (for example, the diad corresponding to a comonad, or any idempotent endofunctor) on a topos is again a topos.     

Representation and extension of states on MV-algebras

MV-algebras stand for the many-valued Łukasiewicz logic the same as Boolean algebras for the classical logic. States on MV-algebras were first mentioned [20] in probability theory and later also introduced in effort to capture a notion of `an average truth-value of proposition' [15] in Łukasiewicz many-valued logic. In the presented paper, an integral representation theorem for finitely-additive states on semisimple MV-algebra will be proven. Further, we shall prove extension theorems concerning states defined on sub-MV-algebras and normal partitions of unity generalizing in this way the well-known Horn-Tarski theorem for Boolean algebras. The author gratefully acknowledges the support of grant 201/02/1540 of the Grant Agency of the Czech Republic and the partial support by the project 1M6798555601 of the Ministry of Education, Youth and Sports of the Czech Republic.     

Algebra in a Grothendieck topos: Injectivity in quasi-equational classes

This paper gives a study of injectivity and some related notions in quasi-equational classes of algebras in an arbitrary Grothendieck topos E. The main purpose is to describe the relationship between the class mod Σ of models of a set Σ of quasi-equations in the category of sets Ens and the corresponding class mod(Σ, E) of models of Σ in E with respect to residual smallness, boundedness of essential extensions and injectivity. The basic nature of our results is that, for any given Σ, whatever holds in Ens, concerning these notions, also holds in E. In particular, this substantially improves the earlier results of Howlett [5] regarding the existence of enough injectives in mod(Σ, E).     

Ultrapowers as sheaves on a category of ultrafilters

In the paper we investigate the topos of sheaves on a category of ultrafilters. The category is described with the help of the Rudin-Keisler ordering of ultrafilters. It is shown that the topos is Boolean and two-valued and that the axiom of choice does not hold in it. We prove that the internal logic in the topos does not coincide with that in any of the ultrapowers. We also show that internal set theory, an axiomatic nonstandard set theory, can be modeled in the topos.Mathematics Subject Classification (2000): Primary 03G30, 03C20, Secondary 03E05, 03E70, 03H05The author would like to thank the Mittag-Leffler Institute for partial suport.     

On algebras defined by operations and equations in a topos

A definition of internal family of equations in the algebras for an internal type in an elementary topos with natural numbers object is given. The category of algebras which satisfy an internal family of equations is shown to be monadic over the base topos and internally cocomplete provided the topos in Grothendieck over a topos with the internal Axiom of Choice.     

Algebraic Representation Theory of Partial Algebras

. This paper develops an algebraic representation theory of partial algebras. We introduce the notion of irreducibility for representations of a partial algebra, prove necessary and sufficient conditions for a representation to be (a) irreducible or (b) completely reducible, and develop aspects of the decomposition theory of completely reducible representations. An application to boson quantum field theory is furnished.     

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ?-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ?-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ?-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.     

Centrality and connectors in Maltsev categories

We develop a new approach to the classical property of centrality of equivalence relations. The internal notion of connector allows to clarify classical results in Maltsev varieties and to extend them in the more general context of regular Maltsev categories, hence including the important new examples of Maltsev quasivarieties and of topological Maltsev algebras. We also prove that Maltsev categories can be characterized in terms of a property of connectors.     

The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra are completely additive, he showed that the completion of always exists and is unique up to isomorphisms over . Moreover, strictly positive equations are preserved under completions a strictly positive equation that holds in must hold in the completion of . In this paper we extend Monk’s preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given that shows that arbitrary Sahlqvist equations need not be preserved. Received May 3, 1998; accepted in final form October 7, 1998.     

Laminations,or how to build a quantum-logic-valued model of set theory

An explicit construction of the colimit of a filtered diagram in the category of topoi and logical morphisms is given and then used to construct a family of topoi with a fixed Boolean algebra of truth values but with varying amounts of cocompleteness. This same construction, when applied to the diagram of complete Boolean algebras in a quantum logic Q gives a partial topos, a noncategory which is a close to being a model of set theory with algebra of truth values Q as a noncategory can be.     

Lusztig's isomorphism theorem for cellular algebras

In this paper, we first introduce a notion of semisimple system with parameters, then we establish Lusztig's isomorphism theorem for any cellular semisimple system with parameters. As an application, we obtain Lusztig's isomorphism theorem for Ariki-Koike algebras, cyclotomic q-Schur algebras and Birman-Murakami-Wenzl algebras. Second, using the results for certain Ariki-Koike algebras, we prove an analogue of Lusztig's isomorphism theorem for the cyclotomic Hecke algebras of type G(p,p,n) (which are not known to be cellular in general). These generalize earlier results of [G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981) 490-498.] on such isomorphisms for Iwahori-Hecke algebras associated to finite Weyl groups.     

Decomposition theorem for the cd-index of Gorenstein<Superscript>*</Superscript> posets

We prove a decomposition theorem for the cd-index of a Gorenstein^* poset analogous to the decomposition theorem for the intersection cohomology of a toric variety. From this we settle a conjecture of Stanley that the cd-index of Gorenstein* lattices is minimized on Boolean algebras.