Peter Jipsen | Chapman University (original) (raw)

Papers by Peter Jipsen

ACM Transactions on Computational Logic

In this paper we extend the research programme in algebraic proof theory from axiomatic extension... more In this paper we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalise the residuated frames in [34] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi D.LE associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus D.LE, as well as for its extensions with analytic structural rules satisfying certain additional properties.

arXiv (Cornell University), Jul 22, 2020

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residu... more We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective helps us find a necessary and sufficient condition for the interval [0, 1] to be a subalgebra of an involutive residuated lattice. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the s... more It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system

Primitive lattice varieties

International Journal of Algebra and Computation, 2022

A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, ... more A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition (W). For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition (W), then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail (W) also generate primitive varieties. But if I is a (W)-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with I doubled, then [Formula: see text] is never primitive.

General results

Varieties of Lattices, 1992

Amalgamation in lattice varieties

Varieties of Lattices, 1992

Positive MV-algebras are negation-free and implication-free subreducts of MV-algebras. In this co... more Positive MV-algebras are negation-free and implication-free subreducts of MV-algebras. In this contribution we show that a finite axiomatic basis exists for the quasivariety of positive MV-algebras coming from any finitely generated variety of MV-algebras. 1 Positive subreducts of MV-algebras Let MV be the variety of MV-algebras [1] in the language containing all the usual definable operations and constants. Using this signature we denote an MV-algebra M ∈ MV as M = 〈M,⊕,⊙,∨,∧,→,¬, 0, 1〉. An algebra A = 〈A,⊕,⊙,∨,∧, 0, 1〉 is a positive subreduct of M if A is a subreduct of M. Definition 1. Let F = {⊕,⊙,∨,∧, 0, 1} be a set of function symbols, where ⊕,⊙,∨,∧ are interpreted as binary operations and 0, 1 as constants. An algebra P of type F is a positive MV-algebra if it is isomorphic to a positive subreduct of some MV-algebra. Clearly, every MV-algebra gives rise to a positive MV-algebra and every bounded distributive lattice is a positive MV-algebra. In fact, positive MV-algebras are ...

Unary-Determined Distributive \ell -magmas and Bunched Implication Algebras

Relational and Algebraic Methods in Computer Science, 2021

Algebra universalis, 2017

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek ... more We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.

Special Issue: Recent Developments related to Residuated Lattices and Substructural Logics Preface

An orderly algorithm to enumerate finite (semi) modular lattices

An online database of ordered algebraic structures

Ordered algebraic structures in the open-source mathematics system Sage

The lattice of varieties generated by residuated lattices of size up to 5

Complex algebras of rectangular bands

ABSTRACT . This note proves that the variety generated by complex algebras of rectangular bands i... more ABSTRACT . This note proves that the variety generated by complex algebras of rectangular bands is finitely based. This is in contrast to the variety generated by complex algebras of all semigroups. Given a semigroup (S; Delta), the complex algebra of S is Cm(S) = (P(S); [; "; ; ;; S; Delta), where XY = fxy : x 2 X; y 2 Y g ( X = SnX, and in the abstract setting ; and S are denoted by 0 and 1). For a class K of semigroups, CmK = fCm(S) : S 2 Kg. The variety generated by CmK is denoted by VCmK. The variety RB of rectangular bands is the class of semigroups defined by the identity xyx = x. In the lattice of semigroup varieties it covers the two varieties Lz and Rz of left-zero and right-zero semigroups that are defined by xy = x and xy = y respectively. Bergman [2] showed that VCmLz and VCmRz are finitely based. An equational basis for VCmLz is given by equations for the Boolean part, together with x(y [ z) = xy [ xz, (x [ y)z = xz [ yz, 0x = 0 = x0, x(yz) = (xy)z (this much defines the var...

Decision Procedures in Algebra and Logic

Computational Investigations of the Lattice of Lattice Varieties

ACM Transactions on Computational Logic

In this paper we extend the research programme in algebraic proof theory from axiomatic extension... more In this paper we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalise the residuated frames in [34] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi D.LE associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus D.LE, as well as for its extensions with analytic structural rules satisfying certain additional properties.

arXiv (Cornell University), Jul 22, 2020

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residu... more We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective helps us find a necessary and sufficient condition for the interval [0, 1] to be a subalgebra of an involutive residuated lattice. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the s... more It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system

Primitive lattice varieties

International Journal of Algebra and Computation, 2022

A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, ... more A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition (W). For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition (W), then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail (W) also generate primitive varieties. But if I is a (W)-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with I doubled, then [Formula: see text] is never primitive.

General results

Varieties of Lattices, 1992

Amalgamation in lattice varieties

Varieties of Lattices, 1992

Positive MV-algebras are negation-free and implication-free subreducts of MV-algebras. In this co... more Positive MV-algebras are negation-free and implication-free subreducts of MV-algebras. In this contribution we show that a finite axiomatic basis exists for the quasivariety of positive MV-algebras coming from any finitely generated variety of MV-algebras. 1 Positive subreducts of MV-algebras Let MV be the variety of MV-algebras [1] in the language containing all the usual definable operations and constants. Using this signature we denote an MV-algebra M ∈ MV as M = 〈M,⊕,⊙,∨,∧,→,¬, 0, 1〉. An algebra A = 〈A,⊕,⊙,∨,∧, 0, 1〉 is a positive subreduct of M if A is a subreduct of M. Definition 1. Let F = {⊕,⊙,∨,∧, 0, 1} be a set of function symbols, where ⊕,⊙,∨,∧ are interpreted as binary operations and 0, 1 as constants. An algebra P of type F is a positive MV-algebra if it is isomorphic to a positive subreduct of some MV-algebra. Clearly, every MV-algebra gives rise to a positive MV-algebra and every bounded distributive lattice is a positive MV-algebra. In fact, positive MV-algebras are ...

Unary-Determined Distributive \ell -magmas and Bunched Implication Algebras

Relational and Algebraic Methods in Computer Science, 2021

Algebra universalis, 2017

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek ... more We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.

Special Issue: Recent Developments related to Residuated Lattices and Substructural Logics Preface

An orderly algorithm to enumerate finite (semi) modular lattices

An online database of ordered algebraic structures

Ordered algebraic structures in the open-source mathematics system Sage

The lattice of varieties generated by residuated lattices of size up to 5

Complex algebras of rectangular bands

ABSTRACT . This note proves that the variety generated by complex algebras of rectangular bands i... more ABSTRACT . This note proves that the variety generated by complex algebras of rectangular bands is finitely based. This is in contrast to the variety generated by complex algebras of all semigroups. Given a semigroup (S; Delta), the complex algebra of S is Cm(S) = (P(S); [; "; ; ;; S; Delta), where XY = fxy : x 2 X; y 2 Y g ( X = SnX, and in the abstract setting ; and S are denoted by 0 and 1). For a class K of semigroups, CmK = fCm(S) : S 2 Kg. The variety generated by CmK is denoted by VCmK. The variety RB of rectangular bands is the class of semigroups defined by the identity xyx = x. In the lattice of semigroup varieties it covers the two varieties Lz and Rz of left-zero and right-zero semigroups that are defined by xy = x and xy = y respectively. Bergman [2] showed that VCmLz and VCmRz are finitely based. An equational basis for VCmLz is given by equations for the Boolean part, together with x(y [ z) = xy [ xz, (x [ y)z = xz [ yz, 0x = 0 = x0, x(yz) = (xy)z (this much defines the var...

Decision Procedures in Algebra and Logic

Computational Investigations of the Lattice of Lattice Varieties