# Peter Jipsen Chapman University, Orange

## Algebra, Logic and Foundations of Mathematics, Theory of Computation

## Publications

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**ABSTRACT:**Lattices have many applications in mathematics and logic, in which they occur together with additional operations. For example, in applications of Hilbert spaces, one is often con- cerned with the lattice of closed subspaces of a fixed space. This lattice is not distributive, but there is an operation taking a given subspace to its orthogonal subspace. More gen- erally, ortholattices are lattices with a unary operation ( )y that is involutive (a = ayy), sends finite joins to meets and for which a and ay are complements. Bounded modal lat- tices (L;_;^; 0; 1; ;2) are models of (not necessarily distributive) modal logic, where and 2 are unary operations that preserve finite join and finite meet, respectively, and represent possible and necessary. Bounded lattice-ordered monoids are bounded lattices with an associative binary operation and an identity element 1. In these examples it is postulated that the additional operations "preserve structure" in various different senses. Orthocomplementation sends finite joins to meets (and finite meets to joins). The modal operators preserve finite joins and finite meets, respectively. Similarly, the monoid oper- ation distribute over finite joins. Bounded residuated lattices are bounded lattice-ordered monoids with two further operationsn, = that interact with via the universally quantified residuation law:Algebra Universalis 05/2014; 71(3). DOI:10.1007/s00012-014-0275-2 · 0.55 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**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 23, 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 $n$. - [Show abstract] [Hide abstract]

**ABSTRACT:**Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how,frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property. We use our results to prove the decidability of the equational and/or universal theory of several vari- eties of residuated lattice-ordered groupoids, including the variety of involutive FL-algebras. Substructural logics and their algebraic formulation as varieties of residuatedTransactions of the American Mathematical Society 01/2013; 365(3):1219-1249. DOI:10.1090/S0002-9947-2012-05573-5 · 1.10 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.Algebra Universalis 01/2013; 69(1). DOI:10.1007/s00012-012-0215-y · 0.55 Impact Factor -
##### Article: Special Issue: Recent Developments related to Residuated Lattices and Substructural Logics Preface

Studia Logica 12/2012; 100(6):1059-1062. DOI:10.1007/s11225-012-9461-4 · 0.33 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**It is proved that any lattice-ordered pregroup that satisfies an identity of the form x ll···l = x rr···r (for the same number of l,r -operations on each side) has a lattice reduct that is distributive. It follows that every such ℓ-pregroup is embedded in an ℓ-pregroup of residuated and dually residuated maps on a chain.Algebra Universalis 10/2012; 68(1-2). DOI:10.1007/s00012-012-0199-7 · 0.55 Impact Factor -
##### Article: Preface

Studia Logica 01/2012; 100(6-6). · 0.33 Impact Factor -
##### Chapter: Background Material

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**ABSTRACT:**This chapter serves the rest of the book: all later chapters presuppose it. It introduces the calculus of binary relations, and relates it to basic concepts and results from lattice theory, universal algebra, category theory and logic. It also fixes the notation and terminology to be used in the rest of the book. Our aim here is to write in a way accessible to readers who desire a gentle introduction to the subject of relational methods. Other readers may prefer to go on to further chapters, only referring back to Chapt. 1 as needed.07/2011: pages 1-21; - [Show abstract] [Hide abstract]

**ABSTRACT:**The poset product construction is used to derive embedding theorems for several classes of generalized basic logic algebras (GBL-algebras). In particular it is shown that every n-potent GBL-algebra is embedded in a poset product of finite n-potent MV-chains, and every normal GBL-algebra is embedded in a poset product of totally ordered GMV-algebras. Representable normal GBL-algebras have poset product embeddings where the poset is a root system. We also give a Conrad–Harvey–Holland-style embedding theorem for commutative GBL-algebras, where the poset factors are the real numbers extended with −∞. Finally, an explicit construction of a generic commutative GBL-algebra is given, and it is shown that every normal GBL-algebra embeds in the conucleus image of a GMV-algebra.Journal of Pure and Applied Algebra 09/2010; 214(9):1559-1575. DOI:10.1016/j.jpaa.2009.11.015 · 0.58 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**It is shown that the Boolean center of complemented elements in a bounded in- tegral 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.Annals of Pure and Applied Logic 11/2009; 161(2):228-234. DOI:10.1016/j.apal.2009.05.005 · 0.45 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Generalized basic logic algebras (GBL-algebras for short) have been introduced in [JT02] as a generalization of Hájek’s BL-algebras, and constitute a bridge between algebraic logic and ℓ-groups. In this paper we investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum decomposition theorem. This result allows us to derive many interesting consequences, such as the decidability of the universal theory of commutative GBL-algebras, the fact that n-potent GBL-algebras are commutative, and a representation theorem for finite GBL-algebras as poset sums of GMV-algebras, a result which generalizes Di Nola and Lettieri’s [DL03] representation of finite BL-algebras.Algebra Universalis 05/2009; 60(4):381-404. DOI:10.1007/s00012-009-2106-4 · 0.55 Impact Factor -
##### Conference Paper: Domain and Antidomain Semigroups.

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**ABSTRACT:**We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semi- groups and dynamic predicate logic.Relations and Kleene Algebra in Computer Science, 11th International Conference on Relational Methods in Computer Science, RelMiCS 2009, and 6th International Conference on Applications of Kleene Algebra, AKA 2009, Doha, Qatar, November 1-5, 2009. Proceedings; 01/2009 -
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**ABSTRACT:**Petr Hajek identified the logic BL, that was later shown to be the logic of continuous t-norms on the unit interval, and defined the corresponding algebraic models, BL-algebras, in the context of residuated lattices. The defining characteristics of BL-algebras are representability and divisibility. In this short note we survey re- cent developments in the study of divisible residuated lattices and attribute the inspiration for this investigation to Petr Hajek.Witnessed years---essays in honour of Petr Hájek, 01/2009: pages 303-327; Coll. Publ., London. -
##### Article: Topological duality and lattice expansions, I: A topological construction of canonical extensions

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**ABSTRACT:**The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone's duality for distributive lattices rather than Priestley's. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper.Algebra Universalis 01/2009; 71(2). DOI:10.1007/s00012-014-0267-2 · 0.55 Impact Factor -
##### Conference Paper: The Structure of the One-Generated Free Domain Semiring.

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**ABSTRACT:**This note gives an explicit construction of the one-generated free domain semiring. In particular it is proved that the elements can be represented uniquely by finite antichains in the poset of finite strictly decreasing sequences of nonnegative integers. It is also shown that this domain semiring can be represented by sets of binary relations with union, composition and relational domain as operations.Relations and Kleene Algebra in Computer Science, 10th International Conference on Relational Methods in Computer Science, and 5th International Conference on Applications of Kleene Algebra, RelMiCS/AKA 2008, Frauenwörth, Germany, April 7-11, 2008. Proceedings; 01/2008 -
- 01/2007: pages xxii+509; Elsevier B. V., Amsterdam., ISBN: 9780444521415
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**ABSTRACT:**A generalized BL - algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities xL y = ((xL y)/y)y = y(y\(xL y)) x\Lambda y = ((x\Lambda y)/y)y = y(y\backslash (x\Lambda y)) . It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements. Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of l {\ell } -group varieties into the lattice of varieties of integral GBLalgebras. The results of this paper also apply to pseudo-BL algebras.Algebra Universalis 01/2006; 55(2):227-238. DOI:10.1007/s00012-006-1960-6 · 0.55 Impact Factor -
##### Article: Minimal Expansions of Semilattices.

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**ABSTRACT:**We determine the minimal extension of the sequence h0,1,1,...,1,2i. This completes and extends the work of K. M. Koh, started in 1970, and solves Problem 15 in the survey on pn- sequences and free spectra (GK92). The results involve the inves- tigation of some minimal expansions of semilattices.International Journal of Algebra and Computation 08/2004; 14(4):465-477. DOI:10.1142/S0218196704001852 · 0.44 Impact Factor

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