We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we (...) derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices. (shrink)

We present a manuscript of Paul Lorenzen that provides a proof of consistency for elementary number theory as an application of the construction of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also compare this proof to Gentzen's and Novikov's, and provide a translation of the manuscript.

It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element |$0$|⁠, then the relative pseudocomplement with respect to |$0$| is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet- (...) class='Hi'>semilattice with |$0$| satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation |$x^{0}$| and implication |$x\rightarrow y$| as the set of all maximal elements |$z$| satisfying |$x\wedge z=0$| and as the set of all maximal elements |$z$| satisfying |$x\wedge z\leq y$|⁠, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry |$x$| or to two entries |$x$| and |$y$| belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts. (shrink)

Sahlqvist theory is extended to the fragments of the intuitionistic propositional calculus that include the conjunction connective. This allows us to introduce a Sahlqvist theory of intuitionistic character amenable to arbitrary protoalgebraic deductive systems. As an application, we obtain a Sahlqvist theorem for the fragments of the intuitionistic propositional calculus that include the implication connective and for the extensions of the intuitionistic linear logic.

In this paper we introduce a class of inquisitive Heyting algebras as algebraic structures that are isomorphic to algebras of finite antichains of bounded implicative meet semilattices. It is argued that these structures are suitable for algebraic semantics of inquisitive superintuitionistic logics, i.e. logics of questions based on intuitionistic logic and its extensions. We explain how questions are represented in these structures and provide several alternative characterizations of these algebras. For instance, it is shown that a Heyting algebra is inquisitive (...) if and only if its prime filters and filters generated by sets of prime elements coincide and prime elements are closed under relative pseudocomplement. We prove that the weakest inquisitive superintuitionistic logic is sound with respect to a Heyting algebra iff the algebra is what we call a homomorphic p-image of some inquisitive Heyting algebra. It is also shown that a logic is inquisitive iff its Lindenbaum–Tarski algebra is an inquisitive Heyting algebra. (shrink)

The semilattice relevant logics ∪R, ∪T, ∪RW, and ∪TW are defined by semilattice models in which conjunction and disjunction are interpreted in a natural way. For each of them, there is a cut-free labelled sequent calculus with plural succedents . We prove that these systems are equivalent, with respect to provable formulas, to the restricted systems with single succedents . Moreover, using this equivalence, we give a new Hilbert-style axiomatizations for ∪R and ∪T and prove equivalence between two (...) semantics for the contractionless logics ∪RW and ∪TW. (shrink)

Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On (...) the other hand, we also give a sufficient condition on a, that yields that there is a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of cardinality n, whose Rogers semilattice consists of exactly one element. (shrink)

We look at lower semilattice-ordered residuated semigroups and, in particular, the representable ones, i.e., those that are isomorphic to algebras of binary relations. We will evaluate expressions in representable algebras and give finite axiomatizations for several notions of validity. These results will be applied in the context of substructural logics.

We devise exact conditions under which a join semilattice with a weak contact relation can be semilattice embedded into a Boolean algebra with an overlap contact relation, equivalently, into a distributive lattice with additive contact relation. A similar characterization is proved with respect to Boolean algebras and distributive lattices with weak contact, not necessarily additive, nor overlap.

We consider${\cal S}$, the class of finite semilattices;${\cal T}$, the class of finite treeable semilattices; and${{\cal T}_m}$, the subclass of${\cal T}$which contains trees with branching bounded bym. We prove that${\cal E}{\cal S}$, the class of finite lattices with linear extensions, is a Ramsey class. We calculate Ramsey degrees for structures in${\cal S}$,${\cal T}$, and${{\cal T}_m}$. In addition to this we give a topological interpretation of our results and we apply our result to canonization of linear orderings on finite semilattices. In (...) particular, we give an example of a Fraïssé class${\cal K}$which is not a Hrushovski class, and for which the automorphism group of the Fraïssé limit of${\cal K}$is not extremely amenable but is uniquely ergodic. (shrink)

The operational semantics of Urquhart is a deep and important part of the development of relevant logics. In this paper, I present an overview of work on Urquhart’s operational semantics. I then present the basics of collection frames. Finally, I show how one kind of collection frame, namely, functional set frames, is equivalent to Urquhart’s semilattice semantics.

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well (...) as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets. (shrink)

The duality between congruence lattices of semilattices, and algebraic subsets of an algebraic lattice, is extended to include semilattices with operators. For a set G of operators on a semilattice S, we have \ \cong^{d} {{\rm S}_{p}}}\), where L is the ideal lattice of S, and H is a corresponding set of adjoint maps on L. This duality is used to find some representations of lattices as congruence lattices of semilattices with operators. It is also shown that these congruence (...) lattices satisfy the Jónsson–Kiefer property. (shrink)

In the early 1970s, Alasdair Urquhart proposed a semilattice semantics for relevance logic which he provided with an influential informational interpretation. In this article, I propose a BHK-inspired reinterpretation of the semantics which is related to Kit Fine’s truthmaker semantics. I discuss and compare Urquhart’s and Fine’s semantics and show how simple modifications of Urquhart’s semantics can be used to characterize both full propositional intuitionistic logic and Jankov’s logic. I then present (quasi-)relevant companions for both of these systems. Finally, (...) I provide sound and complete labelled sequent calculi for all of the systems discussed. (shrink)

The paper discusses regularisation of dualities. A given duality between (concrete) categories, e.g. a variety of algebras and a category of representation spaces, is lifted to a duality between the respective categories of semilattice representations in the category of algebras and the category of spaces. In particular, this gives duality for the regularisation of an irregular variety that has a duality. If the type of the variety includes constants, then the regularisation depends critically on the location or absence of (...) constants within the defining identities. The role of schizophrenic objects is discussed, and a number of applications are given. Among these applications are different forms of regularisation of Priestley, Stone and Pontryagin dualities. (shrink)

Boolean algebras are one of the main algebraic tools in the region-based theory of space. T. Ivanova provided strong motivations for the study of mere semilattices with a contact relation. Another significant motivation for considering an even weaker underlying structure comes from event structures with binary conflict in the theory of concurrent systems in computer science. All the above-hinted notions deal with a binary contact relation. Several authors suggested the more general study of n-ary ‘hypercontact’ relations. A similar evolution occurred (...) in the study of the just mentioned event structures in computer science. To unify the above lines of research, in this paper, we study joining semilattices with a hypercontact relation. We provide representation theorems into Boolean algebras. With a single exception, our proofs are choice-free. We also present several examples and problems; in particular, we briefly discuss some connections with event structures and hypergraphs. (shrink)

Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family is limitwise monotonic (l.m.) if every set is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice. The semilattices exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of ‐computable families. We show that (...) every Rogers semilattice of a ‐computable family is isomorphic to some semilattice. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices. In particular, there is an l.m. family S such that is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all ‐computable numberings. We prove that inside this class, the index set of l.m. numberings is ‐complete. (shrink)

The variety of Brouwerian semilattices is amalgamable and locally finite; hence, by well-known results [19], it has a model completion. In this article, we supply a finite and rather simple axiomatization of the model completion.

Modal pseudocomplemented De Morgan algebras were investigated in A. V. Figallo, N. Oliva, A. Ziliani, Modal pseudocomplemented De Morgan algebras, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 53, 1, pp. 65–79, and they constitute a proper subvariety of the variety of pseudocomplemented De Morgan algebras satisfying xΛ* = *))* studied by H. Sankappanavar in 1987. In this paper the study of these algebras is continued. More precisely, new characterizations of mpM-congruences are shown. In particular, one of (...) them is determined by taking into account an implication operation which is defined on these algebras as weak implication. In addition, the finite mpM-algebras were considered and a factorization theorem of them is given. Finally, the structure of the free finitely generated mpM-algebras is obtained and a formula to compute its cardinal number in terms of the number of the free generators is established. For characterization of the finitely-generated free De Morgan algebras, free Boole-De Morgan algebras and free De Morgan quasilattices see: [16, 17, 18]. (shrink)

We prove that the equational theory of a semigroups becomes undecidable if we add a semilattice structure with a ‘touch of symmetric difference’. As a corollary we obtain that the variety of all Boolean algebras with an associative binary operator has a ‘hereditarily’ undecidable equational theory. Our results have implications in logic, e.g. they imply undecidability of modal logics extending the Lambek Calculus and undecidability of Arrow Logics with an associative arrow modality.

An l-hemi-implicative semilattice is an algebra $$\mathbf {A} = $$ A= such that $$$$ is a semilattice with a greatest element 1 and satisfies: for every $$a,b,c\in A$$ a,b,c∈A, $$a\le b\rightarrow c$$ a≤b→c implies $$a\wedge b \le c$$ a∧b≤c and $$a\rightarrow a = 1$$ a→a=1. An l-hemi-implicative semilattice is commutative if if it satisfies that $$a\rightarrow b = b\rightarrow a$$ a→b=b→a for every $$a,b\in A$$ a,b∈A. It is shown that the class of l-hemi-implicative semilattices is a variety. (...) These algebras provide a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an derived operation by $$a \sim b := \wedge $$ a∼b:=∧. Endowing $$$$ with the binary operation $$\sim $$ ∼ the algebra $$$$ results an l-hemi-implicative semilattice, which also satisfies the identity $$a \sim b = b \sim a$$ a∼b=b∼a. In this article, we characterize the commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element. Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices. (shrink)

Contact algebra is one of the main tools in region-based theory of space. In it is generalized by dropping the operation Boolean complement. Furthermore we can generalize contact algebra by dropping also the operation meet. Thus we obtain structures, called contact join-semilattices and structures, called distributive contact join-semilattices. We obtain a set-theoretical representation theorem for CJS and a relational representation theorem for DCJS. As corollaries we get also topological representation theorems. We prove that the universal theory of CJS and of (...) DCJS is the same and is decidable. (shrink)

In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive lattices.

In this paper we study the subdirectly irreducible algebras in the variety of pseudocomplemented De Morgan algebras by means of their De Morgan p‐spaces. We introduce the notion of the body of an algebra and determine when is subdirectly irreducible. As a consequence of this, in the case of pseudocomplemented Kleene algebras, two special subvarieties arise naturally, for which we give explicit identities that characterise them. We also introduce a subvariety of, namely the variety of bundle pseudocomplemented (...) Kleene algebras, fully describe its subvariety lattice and find explicit equational bases for each subvariety. In addition, we study the subvariety of generated by the simple members of, determine the structure of the free algebra over a finite set in this variety and their finite weakly projective algebras. (shrink)

We investigate expansions of Heyting algebras in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators. We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal congruences using a single equation. (...) Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations. (shrink)

In this article, I present a semantically natural conservative extension of Urquhart’s positive semilattice logic with a sort of constructive negation. A subscripted sequent calculus is given for this logic and proofs of its soundness and completeness are sketched. It is shown that the logic lacks the finite model property. I discuss certain questions Urquhart has raised concerning the decision problem for the positive semilattice logic in the context of this logic and pose some problems for further research.

CRS(fc) denotes the variety of commutative residuated semilattice-ordered monoids that satisfy (x ⋀ e)k ≤ (x ⋀ e)k+1. A structural characterization of the subdi-rectly irreducible members of CRS(k) is proved, and is then used to provide a constructive approach to the axiomatization of varieties generated by positive universal subclasses of CRS(k).

In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬ x ∨ ¬¬ x = 1. Some applications are given.

A propositional logic has the variable sharing property if φ → ψ is a theorem only if φ and ψ share some propositional variable. In this note, I prove that positive semilattice relevance logic and its extension with an involution negation have the variable sharing property. Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property (...) for the systems discussed. (shrink)

This paper deals with axiomatization problems for varieties of residuated meet semilattice-ordered monoids. An internal characterization of the finitely subdirectly irreducible RSs is proved, and it is used to investigate the varieties of RSs within which the finitely based subvarieties are closed under finite joins. It is shown that a variety has this closure property if its finitely subdirectly irreducible members form an elementary class. A syntactic characterization of this hypothesis is proved, and examples are discussed.

In this paper we study pseudo-BCH algebras which are semilattices or lattices with respect to the natural relations ≤; we call them pseudo-BCH join-semilattices, pseudo-BCH meet-semilattices and pseudo-BCH lattices, respectively. We prove that the class of all pseudo-BCH join-semilattices is a variety and show that it is weakly regular, arithmetical at 1, and congruence distributive. In addition, we obtain the systems of identities defininig pseudo-BCH meet-semilattices and pseudo-BCH lattices.

We generalize Priestley duality for distributive lattices to a duality for distributive meet-semilattices. On the one hand, our generalized Priestley spaces are easier to work with than Celani’s DS-spaces, and are similar to Hansoul’s Priestley structures. On the other hand, our generalized Priestley morphisms are similar to Celani’s meet-relations and are more general than Hansoul’s morphisms. As a result, our duality extends Hansoul’s duality and is an improvement of Celani’s duality.

We investigate the join semilattice of modal operators on a Boolean algebra B. Furthermore, we consider pairs ⟨f,g⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle f,g \rangle $$\end{document} of modal operators whose supremum is the unary discriminator on B, and study the associated bi-modal algebras.

In this paper we characterize the join irreducible elements of the free algebras on n free generators in the subvarieties of the variety V0 of pseudocomplemented De Morgan algebras satisfying the identity xx′* = ′*.

In this note we shall show that if L is a balanced pseudocomplemented Ockham algebra then the set ${\fancyscript{I}_{k}(L)}$ of kernel ideals of L is a Heyting lattice that is isomorphic to the lattice of congruences on B(L) where ${B(L) = \{x^* | x \in L\}}$ . In particular, we show that ${\fancyscript{I}_{k}(L)}$ is boolean if and only if B(L) is finite, if and only if every kernel ideal of L is principal.

Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, (...) has (up to equivalence) exactly two computable Friedberg numberings ¼ and ν, and ¼ reduces to any computable numbering not equivalent to ν. The question of whether the full statement of Khutoretskii's Theorem fails for families of d.c.e. sets remains open. (shrink)

In this note we shall describe the lattice of the congruences on a balanced Ockham algebra with the pseudocomplementation whose quotient algebras are boolean. This is an extension of the result obtained by Rodrigues and Silva who gave a description of the lattice of congruences on an Ockham algebra whose quotient algebras are boolean.

An endomorphism on an algebra $${\mathcal {A}}$$ is said to be strong if it is compatible with every congruence on $${\mathcal {A}}$$ ; and $${\mathcal {A}}$$ is said to have the strong endomorphism kernel property if every congruence on $${\mathcal {A}}$$, other than the universal congruence, is the kernel of a strong endomorphism on $${\mathcal {A}}$$. Here we characterise the structure of Ockham algebras with balanced pseudocomplementation those that have this property via Priestley duality.

In our paper, monadic modal pseudocomplemented De Morgan algebras are considered following Halmos’ studies on monadic Boolean algebras. Hence, their topological representation theory is used successfully. Lattice congruences of an mmpM is characterized and the variety of mmpMs is proven semisimple via topological representation. Furthermore and among other things, the poset of principal congruences is investigated and proven to be a Boolean algebra; therefore, every principal congruence is a Boolean congruence. All these conclusions contrast sharply with known results for (...) monadic De Morgan algebras. Finally, we show that the above results for mmpM are verified for monadic tetravalent modal algebras. (shrink)

In this paper we prove that the free pseudocomplemented residuated lattices are decomposable if and only if they are Stone, i.e., if and only if they satisfy the identity ¬x ∨ ¬¬x = 1. Some applications are given.

Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same (...) consequence relations for a given spi-logic. (shrink)