Research

Research Interests   Publications Presentations   Other Activities  CV

 

Research Interests

 

Universal Algebra, Ordered Algebraic Structres and Algebraic Logic.
 

 

Publications

 

Book.

 

1.     Residuated Lattices: an algebraic glimpse at substructural logics, with P. Jipsen, T. Kowalski and H. Ono, Studies in Logics and the Foundations of Mathematics, Elsevier, pp. 509+ (2007). (web site)

 

 

Refereed papers.

 

  1. The undecidability of the word problem for distributive residuated lattices, Ordered Algebraic Structures (J. Martinez, ed.), Kluwer Academic Publishers, Dordrecht, 2002, 231-243.  (pdf)

 

  1. Cancellative residuated lattices,  with P. Bahls, J. Cole, P. Jipsen and C. Tsinakis, Algebra Universalis 50 (2003), no. 1, 83-106. (pdf)

 

  1. Equational bases for joins of residuated-lattice varieties, Studia Logica 76(2) (2004), 227-240. (pdf)

 

  1. Adding involution to residuated structures, with J. Raftery, Studia Logica 77(2) (2004), 181-207. (pdf)

 

  1. Generalized MV-algebras, with C. Tsinakis, Journal of Algebra 283(1) (2005), 254-291. (pdf)

 

  1. Minimal varieties of residuated lattices, Algebra Universalis 52(2) (2005), 215-239. (pdf)

 

  1. Algebraization, paremetrized local deduction theorem and interpolation for substructural logics over FL, with H. Ono, Studia Logica 83 (2006), 279-308. (pdf)

 

  1. Glivenko theorems for substructural logics over FL, with H. Ono, Journal of Symbolic Logic 71(4) (2006), 1353-1384. (pdf)

 

  1. Irreducible residuated semilattices and finitely based varieties, with J. Olson and J. Raftery, Reports on Mathematical Logic 43 (2008), 85-108. (pdf)

 

  1. From axioms to analytic rules in nonclassical logics, with A. Ciabattoni and K. Terui, Proceedings of LICS'08, 229-240, 2008. (pdf)

 

  1. Equivalence of closure operators: an order-theoretic and categorical perspective, with C. Tsinakis, Journal of Symbolic Logic 74(3) (2009), 780-810. (pdf)

 

  1. A survey of GBL-algebras (invited book chapter), with P. Jipsen, Tributes vol.10 College Publications (2009), 303-328. (pdf)

 

  1. Cut elimination and strong separation for substructural logics: an algebraic approach, with H. Ono, Annals of Pure and Applied Logic 161 (2010), no. 9, 1097--1133. (pdf)

 

  1. Generalized ordinal sums and translations, Logic Journal of the IGPL (2011) 19 (3), 455-466. (pdf)

 

  1. MacNeille completions of FL-algebras, with A. Ciabattoni and K. Terui, Algebra Universalis 66(4) (2011), 405-420. (pdf)

 

  1. Algebraic proof theory for substructural logics: cut-elimination and completions, with A. Ciabattoni and K. Terui, Annals of Pure and Applied Logic 163(3) (2012), 266-290. (pdf)

 

  1. A category equivalence for odd Sugihara monoids and its applications, with J. Raftery, Journal of Pure and Applied Algebra 216 (2012), 2177-2192. (pdf)

 

  1. Periodic lattice-ordered pregroups are distributive, with P. Jipsen, Algebra Universalis 68(1-2) (2012), 145-150. (pdf)

 

  1. Residuated frames with applications to decidability, with P. Jipsen, Transactions of the AMS 365(3) (2013), 1219-1249. (pdf)

 

  1. Relation algebras as expanded FL-algebras, with P. Jipsen, Algebra Universalis 69(1) (2013), 1-12. (pdf)

 

  1. Cayley’s and Holland’s theorems for idempotent semirings and their applications to residuated lattices, with R. Horcik,Semigroup Forum 87(3) (2013), 569-589. (pdf)

 

  1. Idempotent residuated structures: some category equivalences and their applications, with J. Raftery, Transactions of the AMS 367(5) (2015), 3189-3223.

 

  1. The finite embeddability property for non-commutative knotted extensions of RL, with Riquelmi Cardona, Internat. J. of Algebra and Computation 25(3) (2015), 349–379.

 

  1. Modules over Quantaloids: Applications to the Isomorphism Problem in Algebraic Logic and $\pi$-institutions, with Jose Gil-Ferez, Journal of Pure and Applied Algebra 221(1) (2016), 1-24. (pdf)

 

  1. Proof Theory for lattice-ordered groups, with G. Metcalfe, Annals of Pure and Applied Logic, 167(8) (2016), 707-724. (pdf)

 

  1. Algebraic proof theory for substructural logics: hypersequents, with A. Ciabattoni and K. Terui, Annals of Pure and Applied Logic 168(3) (2017), 693-737. (pdf)

 

  1. Canonical formulas for k-potent commutative intergral residuated lattices, with N. Bezhanishvili and L. Spada, Algebra Universalis 77(3) (2017), 321-343. (pdf)

 

  1. Distributive residuated frames and generalized bunched implication algebras, with P. Jipsen, Algebra Universalis 78(3) (2017), 303-336. (pdf)

 

  1. The FEP for some varieties of fully-distributive knotted residuated lattices, with R. Cardona, Algebra Universalis 78(3) (2017), 363-376. (pdf)

 

  1. Categories of models of R-mingle,­ with W. Fussner, Annals of Pure and Applied Logic 170(10) (2019), 1188-1242. (pdf)

 

  1. The structure of generalized BI-algebras and weakening relation algebras, with P. Jipsen, Algebra Universalis 81(3) (2020) paper #35, 35pp. (pdf)

 

  1. Weakening relation algebras and FL2 algebras, with P. Jipsen, Lecture Notes in Computer Science 12062 (2020), 117-133. (pdf)

 

  1. Lattice-ordered pregroups are semi-distributive, with P. Jipsen, M. Kinyon, and A. Prenosil, Algebra Universalis 82(1) (2021), paper #16, 6pp. (pdf)

 

  1. Theorems of alternatives for substructural logics, with A. Colacito and G. Metcalfe, Outstanding Contributions to Logic, 21: Arnon Avron on semantics and Proof theory of nonclassical logics (2021), 91-105. (pdf)

 

  1. Densification via polynomial extensions, with R. Horcik, Journal of Pure and Applied Algebra 222(3) (2022), paper #106852, 26pp. (pdf)

 

  1. From distributive l-monoids to l-groups, and back again, with A. Colacito, G. Metcalfe, S. Santchi, Journal of Algebra 601 (2022), 129–148. (pdf)

 

  1. Twist structures and Nelson conuclei, with M. Busaniche and M. Marcos, Studia Logica 110(4) (2022), 949-987. (pdf)

 

  1. Most simple extensions of FLe are undecidabe, with G. St. John, Journal of Symbolic Logic 87(3) (2022), 1156-1200. (pdf)

 

  1. Complemented MacNeille completions and algebras of fractions, with A. Prenosil, Journal of Algebra 623 (2023), 288-357. (pdf)

 

  1. Gluings in residuated lattices, with S. Ugolini, Order 40 (2023), 623–664. (pdf)

 

  1. Distributive ℓ-pregroups: generation and decidability, with Isis Gallardo, Journal of Algebra 648 (2024), 9-35.

 

  1. Semiconic Idempotent Logic I: Structure and Local Deduction Theorems, with W. Fussner, Annals of Pure and Applied Logic 175(7) (2024), paper no.103443, 30 pp.

 

  1. Decidability of lattice equations, Studia Logica 112(3) (2024), 607-610.

 

  1. Unilinear residuated lattices: axiomatization, varieties and FEP, with Xiao Zhuang, Algebra Univeralis 85(3) (2024), paper no. 30, 33 pp.

 

  1. Almost minimal varieties of commutative integral residuated lattices, with P. Agliano and M. Marcos, Int. J. Alg. Comput. 34(5) (2024), 807-836.

 

  Submitted papers.

 

 

Non-refereed conference proceedings.

 

·         Involutive commutative distributive residuated lattices, proceedings of the 37th Mathematical Logic Group Meeting, Shizuoka, 2003.

·         Extensions of Glivenko's theorem to non-commutative substructural logics, proceedings of the 39th Mathematical Logic Group Meeting, Gamagori, 2005.

 

Theses.

 

·         Varieties of residuated lattices, Ph.D. thesis, Department of Mathematics, Vanderbilt Universisty, 2003. (pdf)

·         Selected topics on residuated lattices, Qualifying paper, Department of Mathematics, Vanderbilt University, 2000. (pdf)


Edited volumes

 

 

Educational publication.

From here to infinity: a foundation for Calculus, with J. Staples et al., Tomson Learning Custom Publishing, 2001.

 

Conference Presentations and Seminar Talks

 

Lattice-ordered groups and pregroups, Chapman, December 2023.

Lattice-ordered pregroups: generation and decidability, Siena Algebra Week, July, 2023.

Heyting residuated lattices and weakening relations (invited talk), AMS sectional meeting, El Paso, Texas, September 2022.

Idempotent residuated chains, BLAST, Chapman, August 2022.

Generalized bunched implication logic, PLS, Volos, July 2022.

Lattice-ordered groups and monoids, TACL, Coimbra, June 2022.

Theorems of the alternatives (invited talk), Seminario de Logica algebraica, Argentina, online, April 2022.

The structure of idempotent residuated chains (invited talk), Ordered Algebras and Logic, Les Diablerets, Switzerland. March 2022.

Interpolation and Beth definability for conic idempotent Full Lambek calculus (invited talk), CLAM Latin American Congress of Mathematicians, Special Session on Logic, Montevideo, Uruguay, September 2021.

Lattice-ordered groups and monoids (invited talk), ACaCS conference on semigroups Nis, Serbia, September 2021.

Twist structures and Nelson conuclei, BLAST online at NMSU, Las Cruces NM, June 2021.

Almost minimal varieties of commutative integral residuated lattices, NMSU Algebra and Logic seminar, May 2021.

Nelson conucleus algebras as semantics for paraconsistent logics and their twist-product representation, Alophis online seminar, Cagliari, Italy, May 2021.

Distributive lattice-ordered monoids, Algebra and Coalgebra online seminar, Amsterdam, May 2021.

Amalgamation for conic idempotent residuated lattices, Algebra and Logic online seminar, Boulder, CO, April 2021.

Strong amalgamation for a variety of idempotent semiconic residuated lattices, Nonclassical Logic Webinar, Zoom, July 2020.

Lattice-ordered groups and their subreducts, Shanks Workshop on Ordered Algebras and Logic, Vanderbilt University, Nashville, March 2020.

Inverse-free subreducts of lattice-ordered groups, Special AMS session on Algebras and Algorithms, Joint Mathematical Meetings, Denver, January 2020.

Heyting residuated lattices, Algebra and Logic seminar, Vanderbilt University, Nashville, November 2019.

GBI-algebras and weakening relations (invited talk), Logic Workshop, Amsterdam, Holland, October 2019.

Undecidability for some varieties of commutative residuated lattices, TACL, Nice, France, June 2019.

Generalized bunch implication algebras, BLAST, UC Boulder, May 2019.

Decidability an undecidability for residuated lattices, Algebra and Logic Seminar, Vanderbilt University, April 2019.

Skew reflections, ASL winter meeting, Denver, February, 2019.

Is there a useful duality for residuate lattices?(invited workshop talk), SYSMICS, Chapman University, Orange, September 2018.

Embedding lattice-ordered bi-monoids in involutive commutative residuated lattices (invited talk), LATD, Bern, Switzerland, August 2018.

Skew rotations, AbubL, Cagliari, Italy, June 2018.

Undecidability for some varieties of commutative residuated lattices (invited talk), BLAST, Vanderbilt University, Nashville, August 2017.

Generalized bunched implication algebras, TACL, Prague, June 2017.

Residuated frames and applications (invited talk), WoLC’16, Guanzhou, China, December 2016.

The FEP for residuated lattices via local finiteness of the monoid reducts, Lattices Algebras and Truth Degrees, Wits University, South Africa, June 2016.

Canonical formulas for k-potent commutative residuated lattces, Order Algebras and Logic, Vanderbilt Univesity, Nashville, February 2016.

The FEP for some noncommutative varieties of fully distributive residuated lattices, TACL 2015, Ischia, Italy, June 2015.

Densification via polynomials (invited talk), Shanks workshop in ordered algebras and logic, March 2015.

From display logic to nested sequents via residuated frames (invited talk), Vienna Summer of Logic, GetFun, July 2014.

Embeddings into BiFL-algebras and conservativity, Vienna Summer of Logic, LATD, July 2014.

Proof theory for lattice-ordered groups (invited talk), Brazilian Logic Conference, Petropolis, Brazi, April 2014.

A survey of residuated frames with connections to semirings and to display logic, Prague seminar of substructural logics, March 2014.

The finite embeddability property for non-commutative knotted extension of RL, Colloquium talk at the Institute of CS, Czech Academy of Sciences, March 2014.

Developments on higher levels of the substructural hierarchy, BLAST, Chapman University, Orange, August 2013.

Cayley and Holland Theorems for Residuated Lattices, TACL, Nashville, July 2013.

Hyper-residuated frames (invited talk), ALCOP, Utrecht, Holland. April, 2013.

Distributive integral residuated lattices have the FEP (invited talk), Special Session on Algebras, Lattice and Varieties, AMS Sectional meeting, Boulder, April 2013.

The finite embeddability property for varieties of distributive, integral residuated lattices, Logic, Algebra and Truth Degrees, Kanazawa, Japan, September 2012.

Rediscovering Structural Proof Theory: an algebraic account (invited seminar talk), Multi-dimension seminar, Japan Advanced Institute of Science and Technology, September 2012.

A categorical equivalence of varieties and the Beth definability property, (invited talk), 15th Latin American Symposium on Mathematical Logic, Colombia, scheduled for June 2012.

Algebraic Logic: Algebraic Proof Theory, (invited tutorial),15th Latin American Symposium on Mathematical Logic, Colombia, scheduled for June 2012.

Modules over quantaloids and the isomorphism problem in abstract algebraic logic (invited talk), Symposium on Fuzzy Mathematics, Kent State University and Youngstown State University, Ohio, October 2011.

Residuated lattices in algebra and logic (invited colloquium talk), Colloquium, Department of Mathematical Sciences, Kent State University, Ohio, October 2011.

Relativizing the substructural hierarchy (invited plenary talk), Topology, Algebra and Categories in Logic, Marseilles, France, July 2011.

Interpolation and Amalgamation for Substructural Logics and Residuated Lattices, 8th Panhellenic Logic Symposium, Ioannina, Greece, July 2011.

Beth definability for relevance logic with mingle (invited plenary talk), Second International Conference on Order, Algebra, and Logics: honoring R. McKenzie, H. Ono, and A. Wroński, Krakow, Poland, June 2011.

Holland-Type Embeddings for Residuated Lattices (invited talk), Algebra and Proof Theory amplified by Frames and Category Theory, Vanderbilt University, Nashville,  March 2011.

Remarks on l-groups and abelian l-groups: preliminary report (invited talk), l-Groups and MV-Algebras: Interaction and Impact on Algebras of Logic, Buenos Aires, Argentina, October 2010.

Tutorial on Algebraic Proof Theory (week-long invited tutorial), University of Milan, Italy, July 2010.

Dedekind-MacNeille completions of residuated lattices, BLAST 2010 conference on Algebraic Logic, Set Theory and Universal Algebra, University of Colorado, Boulder, June 2010.

The Isomorphism Problem for modules over quantaloids, World Congress and School on Universal Logic III, Lisbon, Portugal, April 2010.

Residuated lattices with applications to logic, 31st Linz Seminar on Fuzzy Set Theory. (Topic: Lattice-Based Logic and its Applications.) Linz, Austria, February 2010.

Applications of residuated frames to algebra and logic, BLAST 2009 conference on Algebraic Logic, Set Theory and Universal Algebra, New Mexico State University, Las CrucesAugust 2009.

Algebraic Proof Theory (invited plenary talk), 7th Panhellenic Logic Symposium, Patras, Greece, July 2009.

Cut elimination for distributive substructural logics, Topology, Algebra and Categories in Logic, Amsterdam, July 2009.

Sequents, Hypersequents and beyond (invited talk), Gentzen systems and Beyond workshop associated with the TABLEAUX 2009 conference, Oslo, July 2009. 

Non-associative residuated lattices, 2nd Mile Hight Conference on Nonassociative Mathematics, University of Denver, June 2009.

An N2-variety of residuated lattices that is not closed under completions (invited talk), and 12th Annual ORD/OAL, Bowling Green State University, June 2009.

MV-algebras and l-groups as residuated lattices (invited week-long series of lectures), Workshop on lattice-ordered groups and MV-algebras, Bowling Green State University, June 2009.

Skew Relation Algebras (invited talk), Ordered Groups in Logic, Vanderbilt University, March 2009.

Generalized ordinal sums and translations (selected 40-min talk), Logic, Algebra and Truth Degrees, Siena, Italy, September 2008.

Residuated Lattices – a tutorial (invited 3-hour tutorial), Summer school on General Algebra and Ordered Sets, Trest, Czech Republic, September 2008.

The finite embeddability property for subvarieties of integral residuated lattices, BLAST 2008 conference on Algebraic Logic, Set Theory and Universal Algebra, University of Denver, August 2008.

On the substructural hierarchy (invited plenary talk), International Conference on Residuated Structures: Algebra and Logic, Buenos Aires, Argentina, April 2008.

Axiomatizations for intersections of substructural logics (invited 50-min talk), Shanks workshop on Proof Theory and Algebra, Vanderbilt University, March 2008.

Structural rules in FL: expressive power and cut elimination, Algebraic and Topological Methods in Non-classical Logics III, University of Oxford, UK, August 2007.

The variety of involutive residuated lattices is generated by its finite members (invited plenary talk), International conference on Order, Algebra and Logics, Vanderbilt University, USA, June 2007.

Algebraic cut elimination (invited seminar talk), Logic Summer School, Australian National University, Canberra, Australia, December 2006.

Residuated frames (invited talk), Workshop on Algebra and Substructural Logics III, Krakow, Poland, November 2006.

Cut elimination for substructural logics from a semantical perspective, Colloquium talk, Vienna University of Technology, November 2006.

Semantics for substructural logics, Colloquium talk, University of Milan, Italy, October 2006.

Decidability for involutive residuated lattices, Colloquium talk, Univesity of Szeged, Hungary, October 2006.

Glivenko theorems for substructural logics, Logic Colloquium 2006, Association for Symbolic Logic European Summer Meeting, Nijmegen, Netherlands, scheduled for July 2006.

Lack of structural rules and algebraic semantics, Ordered structures in many-valued logics, Massa Lubrense, Sorrento, Italy, May 2006.

Algebraic methods in substructural logics, Logic Colloquium 2005, Association for Symbolic Logic European Summer Meeting, Athens, Greece, July 2005.

Equivalence of consequence relations: an order-theoretic and categorical perspective, Algebraic and topological methods in non-classical logics II, Barcelona, Spain, June 2005.

Non-associative residuated lattices, AMS sectional meeting, special session on Universal Algebra and Lattice Theory, Vanderbilt University, October 2004.

Translations in substructural logics, Logic Colloquium 2004, ASL European Summer Meeting, Torino, Italy, July 2004.

Non-associative substructural logics: algebraization, cut elimination and separation, Conference on Residuated Structures and Many-Valued Logics, Patras University, Greece, June 2004.

Residuated lattices – an introduction, Department of Mathematics, Aristotle University of Thessaloniki, Greece, May 2004.

Involutive commutative distributive residuated lattices, 37th Mathematical Logic Group Meeting, Shizuoka, Japan, December 2003.

The join of finitely many varieties of residuated lattices (invited talk), Annual Meeting of the Association for Symbolic Logic, special session on Algebraic Logic and Universal Algebra, University of Illinois at Chicago, June 2003.

Decision problems for commutative distributive residuated lattices, Colloquium talk, University of Saskatoon, Canada, May 2003.

Generalized MV-algebras, Conference on Ordered Algebraic Structures, Vanderbilt University, March 2003.

A Priestley duality for bounded distributive residuated lattices, Annual meeting of the American Mathematical Society, special session on The Many Lives of Lattice Theory and the Theory of Ordered Sets, with Connections to Combinatorics, Baltimore, January 2003.

A categorical equivalence for integral GMV-algebras, Workshop on Algebra and Substructural Logics, Japan Advanced Institute of Science and Technology, November 2002.

Integral generalized MV-algebras, Algebra Seminar, Vanderbilt University, October 2002.

Some atomic varieties of residuated lattices, Algebra Seminar, Vanderbilt University, September 2002.

Continuum many idempotent minimal residuated-lattice varieties, International Conference on Modern Algebra in conjunction with the 17th annual Shanks Lectures, Vanderbilt University, May 2002.

Varieties of residuated lattices generated by positive universal classes, Conference on Ordered Algebraic Structures, Vanderbilt University, March 2002.

The undecidability of the word problem for the variety of distributive residuated lattices, Conference on Lattice-Ordered Groups and f-Rings, University of Florida, March 2001.

The subvariety lattice of residuated lattices, Workshop on Ordered Algebraic Structures, Vanderbilt University, March 2000.

 

Other Academic Activities

 

Member of the Editorial Board for the journal Algebra Universalis and the journal Mathematical Logic Quarterly.

Guest Editor for the Studia Logica special issue: Recent Developments related to Residuated Lattices and Substructural Logics.

Guest Editor for the Outstanding Contributions, Trends in Logic special issue: Hiroakira Ono on Algebra and substractaral logics.

 Referee for the journals

·         Proceedings of the AMS

·         Journal of Pure and Applied Algebra

·         Algebra Universalis

·         Order

·         Semigroup Forum.

·         Discussiones Mathematicae: General Algebra and Applications

·         Journal of Symbolic Logic

·         Annals of Pure and Applied Logic

·         Studia Logica

·         Archive for Mathematical Logic

·         Reports on Mathematical Logic

·         Mathematical Logic Quarterly

·         Logic Journal of the Interest Group in Pure and Applied Logics

·         Journal of Logic and Computation

·         Journal of Applied Logic

·         Lecture Notes in Computer Science

·         Information Sciences

·         Soft Computing,

·         Fuzzy Sets and Systems

·         Czechoslovak Mathematical Journal

·         Mathematica Slovaca

·         Mathematica Bohemica

Conference Organization:

Master’s and Ph.D. committees

·         Ph.D. supervisor for Xiao Zhang and Isis Gallardo, current.

·         Ph.D. thesis examiner for Frederik Lauridsen, Amsterdam, Holland, October 2019.

·         Postoctoral mentor for Sara Ugolini, September 2018-present.

·         Ph.D. supervisor for Gavin St John. Decidability and Undecidability in substructural logics. Graduated (PhD), May 2019.

·         Ph.D. supervisor for Wesley D. Fussner. Duality theory for Sugihara monoids  Graduated (PhD), December 2018.

·         Ph.D. thesis examiner of Karel Chvalovsky, Charles University, Czech Republic, May 2015.

·         Ph.D. supervisor for Riquelmi Cardona. The Finite Model Property for Substructural Logics with Exchange. Graduated (PhD) in 2015.

·         Ph.D. committee member for Wilmari Morton, University of Johannesburg, 2012.

·         Ph.D. committee member for Jenya Kirstein, University of Denver, 2012.

·         Ph.D. committee member for Topaz Dent, UC Boulder, 2011.

·         Ph.D. committee member for Jose Jil-Ferez, University of Barcelona, 2009.

·         Examiner of the Ph.D. thesis of Ai-ni Hsieh, University of Natal, South Africa, 2009.

·         Ph.D. committee member for Dan Daly, University of Denver, 2009.

·         Ph.D. committee member for Aditya Nagrath, University of Denver, 2008.

·         Examiner of the Master’s thesis of Ai-ni Hsieh, University of Natal, South Africa, 2005.

 

PhD students

 

·         Riquelmi Cardona (PhD 2015), Professor of Mathematics, University of El Salvador.

·         Wesley Fussner (PhD 2018), Postoctoral Fellow, University of Bern.

·         Gavin St. John (PhD 2019), Postdoctoral Fellow, University of Salerno.

·         Xiao Zhuang (PhD 2023), Remotasks.

·         Isis Gallardo (PhD exp. 2024).

·         Kempton Albee (PhD exp. 2026)

·         Sara Ugolini (Postdoc 2018-2020), AI Research Institute, Barcelona.

 

 

 

Curriculum Vitae


Here is a copy of my CV.

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