Publications

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    Peter Jipsen, Nathan Lawless
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    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 $n$.
    09/2013;
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    Nikolaos Galatos, Peter Jipsen
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    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 residuated
    Transactions of the American Mathematical Society 01/2013; 365(3):1219-1249. · 1.02 Impact Factor
  • Article: Preface
    Studia Logica 01/2012; 100(6):1059-1062. · 0.34 Impact Factor
  • Peter Jipsen, Chris Brink, Gunther Schmidt
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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;
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    P. Jipsen, F. Montagna
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    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 01/2010; 214(9):1559-1575. · 0.53 Impact Factor
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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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    M. ANDREW MOSHIER, PETER JIPSEN
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    ABSTRACT: The two main objectives of this paper are (a) to prove topological duality theorems for semilattices and bounded lattices, and (b) to show that the t opological duality from (a) provides a construction of canonical extensions of bounded lattices. The paper is 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. Regarding objective (a), consider the following simple question: Is there a subcategory of Top that is dually equivalent to Lat? Here, Top is the category of topological spaces and continuous maps and Lat is the category of bounded lattices and lattice homomorphisms. To date, the question has been answered positively either by specializing Lat or by generalizing Top. The earliest examples are of the former sort. Tarski (Tar29) (treated in English, e.g., in (BD74)) showed that every complete atomic Boolean lattice is represented by a powerset. Taking some historical license, we can say this result shows that the category of complete atomic Boolean lattices with complete lat- tice homomorphisms is dually equivalent to the category of discrete topological spaces. Birkhoff (Bir37) showed that every finite distributive latt ice is represented by the lower sets of a finite partial order. Again, we can now say that this s hows that the category of finite distributive lattices is dually equivalent to the cat egory of finite T0 spaces and con- tinuous maps. In the seminal papers, (Sto36, Sto37), Stone generalized Tarski and then Birkhoff, showing that (a) the category of Boolean lattices and lattice homomorphisms is dually equivalent to the category of zero-dimensional, r egular spaces and continuous maps and then (b) the category of distributive lattices and l attice homomorphisms is dually equivalent to the category of spectral spaces and spectral maps. We will describe spectral spaces and spectral maps below. For now, notice that all of these results can be viewed as specializing Lat and obtaining a subcategory of Top. In the case of distributive lattices, the topological category is not full because spectral maps are special continuous maps. As a conceptual bridge, Priestley (Pri70) showed that distributive lattices can also be dually represented in a category of certain topological spa ces augmented with a partial
    01/2009;
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    Nikolaos Galatos, Peter Jipsen
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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.
    01/2009: pages 303-327;
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    Peter Jipsen
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    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.
    Ann. Pure Appl. Logic. 01/2009; 161:228-234.
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    Peter Jipsen
    01/2009;
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    Peter Jipsen, Georg Struth
    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
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    01/2007: pages xxii+509; Elsevier B. V., Amsterdam., ISBN: 9780444521415
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    Peter Jipsen
    01/2007;
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    P. Jipsen, F. Montagna
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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. · 0.45 Impact Factor
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    ABSTRACT: We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17].
    Studia Logica 01/2004; · 0.34 Impact Factor
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    Peter Jipsen
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    ABSTRACT: We define the concepts of representable and abstract sequential Q-algebra, which are generalizations of the (relational) Q-algebras in [10]. Just as in that paper, we then prove that the two concepts coincide. In the following section we recall the concept of observation space and note that all complex algebras of observation spaces are representable sequential algebras. Finally we give an uncountable family of representable sequential algebras that generate distinct minimal varieties (i.e. covers of the variety of one-element algebras).
    Journal on Relational Methods in Computer Science. 01/2004; 1:235-250.
  • Peter Jipsen
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    ABSTRACT: We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.
    Studia Logica 01/2004; 76:291-303. · 0.34 Impact Factor
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    Peter Jipsen, Andrzej Kisielewicz
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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.
    IJAC. 01/2004; 14:465-477.
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    ABSTRACT: Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( l \mathcal{l} -groups). Although cancellative monoids are defined by quasi-equations, the class CanRL \mathcal{CanRL} of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL \mathcal{CanRL} that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to l \mathcal{l} -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of CanRL \mathcal{CanRL} .We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of l \mathcal{l} -groups, hence the latter form a variety, denoted by LG- \mathcal{LG}^- . Furthermore we prove that the map that sends a subvariety of l \mathcal{l} -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of LG \mathcal{LG} to the lattice of subvarieties of LG- \mathcal{LG}^- . Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzies characterization of categorically equivalent varieties.
    Algebra Universalis 11/2003; 50(1):83-106. · 0.45 Impact Factor
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    Peter Jipsen
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    ABSTRACT: The main result is that the variety generated by complex algebras of (commutative) semigroups is not finitely based. It is shown that this variety coincides with the variety generated by complex alge- bras of partial (commutative) semigroups. An example is given of an 8-element commutative Boolean semigroup that is not in this variety, and an analysis of all smaller Boolean semigroups shows that there is no smaller example. However, without associativity the situation is quite dierent: the variety generated by complex algebras of (commutative) binars is finitely based and is equal to the variety of all Boolean algebras with a (commutative) binary operator.
    Relational and Kleene-Algebraic Methods in Computer Science: 7th International Seminar on Relational Methods in Computer Science and 2nd International Workshop on Applications of Kleene Algebra, Bad Malente, Germany, May 12-17, 2003, Revised Selected Papers; 01/2003

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