We study the variety Var() of lattice-ordered monoids generated by the natural numbers. In particular, we show that it contains all 2-generated positively ordered lattice-ordered monoids satisfying appropriate distributive laws. Moreover, we establish that the cancellative totally ordered members of Var() are submonoids of ultrapowers of and can be embedded into ordered fields. In addition, the structure of ultrapowers relevant to the finitely generated case is analyzed. Finally, we provide a complete isomorphy invariant (...) in the two-generated case. (shrink)

We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain (...) various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element. (shrink)

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to (...) any finitely accessible category, any first-order definable class of structures, and any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: [email protected]: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf. (shrink)

The paper deals with involutive FLe-monoids, that is, commutative residuated, partially-ordered monoids with an involutive negation. Involutive FLe-monoids over lattices are exactly involutive FLe-algebras, the algebraic counterparts of the substructural logic IUL. A cone representation is given for conic involutive FLe-monoids, along with a new construction method, called twin-rotation. Some classes of finite involutive FLe-chains are classified by using the notion of rank of involutive FLe-chains, and a kind of duality is developed between positive and non-positive rank algebras. As (...) a side effect, it is shown that the substructural logic IUL plus t ↔ f does not have the finite model property. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics. (shrink)

An integral subresiduated lattice ordered commutative monoid (or integral srl-monoid for short) is a pair \(({\textbf {A}},Q)\) where \({\textbf {A}}=(A,\wedge,\vee,\cdot,1)\) is a lattice ordered commutative monoid, 1 is the greatest element of the lattice \((A,\wedge,\vee )\) and _Q_ is a subalgebra of _A_ such that for each \(a,b\in A\) the set \(\{q \in Q: a \cdot q \le b\}\) has maximum, which will be denoted by \(a\rightarrow b\). The integral srl-monoids can be (...) regarded as algebras \((A,\wedge,\vee,\cdot,\rightarrow,1)\) of type (2, 2, 2, 2, 0). Furthermore, this class of algebras is a variety which properly contains the varieties of integral commutative residuated lattices and subresiduated lattices respectively. In this paper we study the quasivariety of \(\{\wedge,\cdot,\rightarrow,1\}\) -subreducts of integral srl-monoids, which will be denoted by \(\mathsf {SR^s}\). In particular, we show that \(\mathsf {SR^s}\) is a variety. We also characterize simple and subdirectly irreducible algebras of \(\mathsf {SR^s}\) respectively. Finally, through a Hilbert style system, we present a logic which has as algebraic semantics the variety \(\mathsf {SR^s}\) and we apply this result in order to present an expansion of the previous logic which has as algebraic semantics the variety of integral srl-monoids. (shrink)

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics.

In any variety of bounded integral residuated lattice-ordered commutative monoids the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the (...) relationship with the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties k>0 defined in Cignoli and Torrens. (shrink)

In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

Special groups [M. Dickmann, F. Miraglia, Special Groups : Boolean-Theoretic Methods in the Theory of Quadratic Forms, Memoirs Amer. Math. Soc., vol. 689, Amer. Math. Soc., Providence, RI, 2000] are a first-order axiomatization of the theory of quadratic forms. In Section 2 we investigate reduced special groups which are a lattice under their natural representation partial order ; we show that this lattice property is preserved under most of the standard constructions on RSGs; in particular finite RSGs and (...) RSGs of finite chain length are lattice ordered. We prove that the lattice property fails for the RSGs of function fields of real algebraic varieties over a uniquely ordered field dense in its real closure, unless their stability index is 1 . We show that Open Problem 1 has a positive answer for the RSG of the field Q . In the final section we explore the meaning of Open Problem 1 for formally real fields, in terms of their orders and real valuations; we introduce the notion of “parameter-rank” of a positive-primitive first-order formula of the language for special groups. (shrink)

We propose a notion of -minimality for partially ordered structures. Then we study -minimal partially ordered structures such that is a Boolean algebra. We prove that they admit prime models over arbitrary subsets and we characterize -categoricity in their setting. Finally, we classify -minimal Boolean algebras as well as -minimal measure spaces.

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson (...) and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

Craig's interpolation theorem fails for the propositional logics E of entailment, R of relevant implication and T of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.

Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, (...) by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions. (shrink)

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian (...) ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. (shrink)

We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. ‘‘Truth values’’ are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] and (...) its equivalent algebraic semantics BAL is definitionally equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of ℓ–groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the co-NP-completeness of the tautology problem of Bal. (shrink)

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras (...) are represented as Esakia products of simple n-potent MV-algebras. (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)

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is (...) of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without (...) involution) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t. (shrink)

The theory of concept lattices is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of vague data—the propositions (...) “object x has attribute y”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better. (shrink)

Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic, using the notion of a first-order hyperdoctrine from categorical logic, to a structure which we will call the hyperdoctrine (...) of mass problems. We study the intermediate logic that the hyperdoctrine of mass problems gives us, and we study the theories of subintervals of the hyperdoctrine of mass problems in an attempt to obtain an analogue of Skvortsova’s result that there is a factor of the Medvedev lattice characterising intuitionistic propositional logic. Finally, we consider Heyting arithmetic in the hyperdoctrine of mass problems and prove an analogue of Tennenbaum’s theorem on computable models of arithmetic. (shrink)

When a set of closed intervals of the reals is partially ordered by decreeing that A (...) all such lattices of finite length. Characterizations are given of when these lattices are modular or distributive, as well as when they are semiorders. The theory is of some interest because the completion by cuts of an interval order is necessarily an interval order lattice. Though it is shown that the completion by cuts of a semiorder need not be a semiorder, necessary and sufficient conditions are given for a lattice of finite length to be isomorphic to the completion by cuts of a semiorder. (shrink)