arXiv subjects

Peter Jipsen

Publications and source records attributed to Peter Jipsen.

21 records · Page 2Linked to original sources

arXiv2018

Nonassociative right hoops

The class of nonassociative right hoops, or narhoops for short, is defined as a subclass of right-residuated magmas, and is shown to be a variety. These algebras generalize both right quasigroups and right hoops, and we characterize the subvarieties in which the operation $x\sqcap y=(x / y)y$ is associative and/or commutative. Narhoops with a left unit are proved to have a top element if and only if $\sqcap$ is commutative, and their congruences are determined by the equivalence class of the left unit. We also show that the four identities defining narhoops are independent.

math.RA↗

arXiv2018

An Algebraic Glimpse at Bunched Implications and Separation Logic

We overview the logic of Bunched Implications (BI) and Separation Logic (SL) from a perspective inspired by Hiroakira Ono's algebraic approach to substructural logics. We propose generalized BI algebras (GBI-algebras) as a common framework for algebras arising via "declarative resource reading", intuitionistic generalizations of relation algebras and arrow logics and the distributive Lambek calculus with intuitionistic implication. Apart from existing models of BI (in particular, heap models and effect algebras), we also cover models arising from weakening relations, formal languages or more fine-grained treatment of labelled trees and semistructured data. After briefly discussing the lattice of subvarieties of GBI, we present a suitable duality for GBI along the lines of Esakia and Priestley and an algebraic proof of cut elimination in the setting of residuated frames of Galatos and Jipsen. We also show how the algebraic approach allows generic results on decidability, both positive and negative ones. In the final part of the paper, we gently introduce the substructural audience to some theory behind state-of-art tools, culminating with an algebraic and proof-theoretic presentation of (bi-)abduction.

cs.LO↗

arXiv2014

Generating all finite modular lattices of a given size

Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18. Here we adapt and improve this algorithm to construct and count modular lattices up to size 24, semimodular lattices up to size 22, and lattices of size 19. We also show that $2^{n-3}$ is a lower bound for the number of nonisomorphic modular lattices of size $n$.

math.CO↗