__Research__

Research
Interests Publications Presentations
Other Activities CV

*Universal Algebra, Ordered Algebraic
Structres and Algebraic Logic.*

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.

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

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

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

* *

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

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

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

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

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

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

* *

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

* *

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

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

*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)

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

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

*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)

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

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

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

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

*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)

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

*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.

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

*Proof Theory for lattice-ordered groups*, with G. Metcalfe, accepted in the Annals of Pure and Applied Logic.

Submitted papers.

·
*Densification via polynomial extensions*,
with R. Horcik, submitted.

·
*Algebraic proof theory for substructural logics: hypersequents,
*with A. Ciabattoni and K. Terui,
submitted.

·
*Decidability of lattice equations, *submitted.

·
*Distributive residuated
frames and generalized bunched implication algebras, *with P. Jipsen, submitted.

·
*The FEP for some varieties of
fully-distributive knotted residuated lattices, *with
R. Cardona, submitted.

·
*Canonical
formulas for k-potent commutative intergral residuated lattices, *with
N. Bezhanishvili and L. Spada,
submitted.

**Non-refereed conference
proceedings.**

*Involutive commutative distributive residuated lattices*,
proceedings of the 37^{th} Mathematical Logic Group Meeting, Shizuoka,
2003.

*Extensions of Glivenko's theorem to non-commutative
substructural logics*, proceedings of the 39^{th} 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)

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**

* *

*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), *15^{th} Latin American Symposium on Mathematical
Logic, Colombia, scheduled for June 2012.

*Algebraic Logic:
Algebraic Proof Theory, (invited tutorial),15 ^{th}* 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, 8 ^{th} 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),* *7^{th} 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,* 2^{nd}
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.

**Member
of the Editorial Board** for the
journal Soft Computing.

** Referee **for the journals

· Journal of Pure and Applied Algebra

· Order

· Discussiones Mathematicae: General Algebra and Applications,

· Annals of Pure and Applied Logic,

· 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,

· Lecture Notes in Computer Science,

·
Czechoslovak Mathematical
Journal,* *

**Conference Organization:**

- Program committee member for LATD, 2016.
- Program committee member for WOLLIC, 2015.
- Program committee member for LATD, 2014.
- Co-chair of TACL, 2013.
- Program committee member for ALCOP (Algebra and Coalgebra Meet Proof Theory), April 2012.
- Program committee member for TACL, 2009.
- Co-chair of BLAST 2008 conference in Algebraic
Logic, Set Theory and Universal Algebra, University of Denver, August
2008.
- Co-organizer of the Workshop in Algebra and Logic, JAIST, March 2007.

**Master’s and Ph.D. committees **

·
Ph.D. committee supervisor of Riquelmi Cardona, University of Denver, 2015.

·
Examiner of the Ph.D. thesis of Karel Chvalovsky, Charles
University, 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.