### Teaching and Research

**Contact:** jharding@nmsu.edu

**Education: **

Ph.D., McMaster University, 1991, Advisor G. Bruns

M. Sc., McMaster University, 1988, Advisor G. Bruns

B.Sc., McMaster University, 1987

**Career history: **

Department Head, New Mexico State University, 2019 –

Professor, New Mexico State University, 2005 –

Associate Professor, New Mexico State University, 1999 – 2005

Assistant Professor, New Mexico State University, 1996 – 1999

Assistant Professor, Brandon University, 1993 –1996

NSERC Postdoctoral Fellow, Vanderbilt University, 1991-1993

**Awards:**

Arts and Sciences Outstanding Faculty Award for NMSU, 2007

D. C. Rousch Award for Excellence in Teaching, received January 2004

International Quantum Structures Association Research Award 2000

SF RTG Grant 2231414: Research Training Group in Logic and its Application, 2023-2028, $1,382,740

US Army Grant W911NF-21-1-0247: Logic and Geometry in Quantum Computing 2021-24, $347,700

Foundational Questions Institute Research Award: Events as Decompositions, 2015-2017, $44,250

**Courses Taught:**

Undergraduate: Calculus I, II, III, Vector Analysis, Differential Equations, Analysis, Algebra, Discrete Mathematics, Linear Algebra, Programming in Pascal, Data Structures, Statistics, Applied Statistics, Survey Sampling, Combinatorics, Great Theorems in Mathematics, Math Appreciation, Quantum Computing.

Graduate: Logic, Lattice Theory, Set Theory, Universal Algebra, Model Theory, Algebra I, II, Linear Algebra, Foundations of Geometry, Measure Theory, Real Analysis.

**Service and Professional Duties:**

Treasurer, International Quantum Structures Association, 2022-

President, International Quantum Structures Association, 2014-2016

Vice President, International Quantum Structures Association, 2016-2018

Councilor, International Quantum Structures Association, 1998-2002, 2006-2010

Editorial Board of Order, 2001 –

Advisory Board Mathematica Slovaca, 2007 –

Chair, Graduate Studies for Mathematical Sciences, NMSU 2005 – 2009

Chair, Graduate recruiting for Mathematical Sciences, NMSU 2011-2013

**Some Recent Talks:**

- Completely Hereditarily Atomic OMLs, SSAOS, Stara Lesna, September 2023: Talk I | Talk II
- Quantum Set Cylindric Algebras, BLAST, Chapman University August 2022.
- Logical aspects of quantum structures, ECONVN (virtual), January 2022.
- Convolution algebras, invited talk at the inaugural meeting of the Malaysian Logic Society (virtual), October 2021.
- The decompositions approach to quantum mechanics, LQCAI (virtual), July 2021.
- Free completely distributive extensions, ILLC workshop (virtual), July 2021.
- Canonical completions, Colloquium talk (virtual), Warsaw, March 2021.
- Decompositions in quantum logic, ARL workshop on higher category theory, Washington, February 2019
- Boolean subalgebras of orthoalgebras, SYSMICS, Orange CA, September 2018
- ESSLLI Lectures on Lattice Theory, Toulouse, July 2017: Lec 1, Lec 2, Lec 3, Lec 4, Lec 5
- The Convolution Algebra, talk at TACL, Prague, June 2017
- An Operational View of Schrodinger’s Equation, talk at the IQSA meeting, Nijmegen, July 2017
- The Type-2 Truth Value Algebra, Talk at AMS meeting, Denver, September 2016
- Automorphisms of Decompositions, talk at the IQSA meeting, Leicester, July 2016
- The Type-2 Truth Value Algebra, Talk at a workshop in Amsterdam, September 2015
- Products or Sums, Talk at Quantum Workshop in Amsterdam, May 2015
- Quantum Structures, Colloquium Iowa State, September 2014
- Topological Boolean Algebras, seminar talk, Iowa State, September 2014
- Projective Bichains, BLAST, Chapman, July 2013
- Type-2 fuzzy sets, Talk at AMS meeting Akron, October 2012
- Modal compact Hausdorff spaces, talk at the Duality Workshop, Oxford, June 2012
- Proximities, Colloquium at UTEP, El Paso, November 2011
- Proximity frames, Talk at BLAST, Lawrence KS, June 2010
- Projective bichains, undelivered talk, planned for BLAST 2010
- Subalgebras of orthomodular lattices, Colloquium at Chapman University, Los Angeles, November, 2010
- Orthomodular structures and categories or Everything old is new again, IQSA meeting, Boston, June 2010
- Logics of Stone spaces, BLAST 2010, Boulder, June 2010
- Daggers, kernels, Baer *-semigroups and orthomodularity, ASL annual meeting, Washington, March 2010
- Orthomodularity in a categorical setting, QPL, Oxford June 2009
- Completions of Ordered Algebraic Structures: A Survey, at UncLog JAIST, Ishikawa Japan, March 2008
- Some Quantum Logic and a few Categories, at the Categorical Quantum Logic Workshop, Oxford, August 2007

**Publications: ** (click on the link for a .pdf file)

- J. Harding and A. Kornell, Completely hereditarily atomic OMLs, submitted to Math. Slovaca.
- J. Harding, J. McDonald and M. Peinado, Monadic ortholattices: completions and duality, submitted to Alg. Univ.
- J. Harding, Quantum cylindric set algebras, submitted to IJTP.
- G. Bezhanishvili and J. Harding, Duality theory for the category of stable compactifications.
*Topology Proc.*61 (2023), 1-30. - Bezhanishvili, J. Harding and P. J. Morandi, Remarks on hyperspaces for Priestley spaces, Theoret. Comput. Sci. 943 (2023), 187-202.
- J. Harding, Quantum monadic algebras, ArXiv 2203.15010v1, 2022.
- J. Harding and Z. Wang, Logical aspects of quantum structures, invited contribution for Financial Econometrics: Bayesian analysis, quantum uncertainty, and related topics, Studies in Systems, Decision and Control, Thatch, Kreinovich, Ha and Trung Eds. Springer, to appear.
- J. Harding and Hung Nguyen, Luders rule and conditional probability for commuting events, 43-56, invited contribution for Prediction and Causality in Econometrics and Related Topics, Studies in Computational Intelligence 983, Ha, Trung, Kreinovich and Thatch Eds. Springer 2022.
- J. Harding, Decompositions in quantum mechanics — an overview, invited contribution for Quantum Computing in Econometrics, Quantum Economics and Related Topics, Studies in Systems, Decision and Control, Sriboonchitta, Kreinovich and Yamaka Eds. Springer. To Appear.
- J. Harding and F. Lauridsen, Hyper-MacNeille completions of Heyting algebras,
*Studia Logica*109 (2021) no. 5, 1119-1157. - M. Cruz-Quinones and J. Harding, Completions of pseudo-ordered sets,
*Order*(2021), DOI: 10.1007/s11083-021-0956-4. - J. Harding and A. J. Lindenhovius, Orthogeometries and AW*-algebras, to appear in
*The Houston J. Math.*. - J. Harding and C. Heunen, Topos quantum theory with short posets,
*Order*38 (2021), no. 1, 111-125. - G. Bezhanishvili and J. Harding, The Fell compactification of a poset, in Statistical and Fuzzy Approaches to Data Processing with Applications to Economics and Other Areas, 31-46, Studies in Computational Intelligence series, vol. 892, 2021 .
- G. Bezhanishvili, J. Harding, and M. Jibladze, Canonical extensions, free completely distributive lattices, and complete retracts,
*Alg. Univer.*82 (2021), no. 4., paper 64. - G. Bezhanishvili and J. Harding, Raney algebras and duality for T_0-spaces,
*Appl. Categ. Structures*29 (2020), no. 6, 963-973. - J. Harding and C. Walker, A topos view of the type-2 fuzzy truth value algebra, in Algebraic Techniques and Their Use in Describing and Processing Uncertainty, 41-54, Studies in Computational Intelligence series, vol. 878, Springer, 2020.
- G. Bezhanishvili, D. Gabelaia, J. Harding, and M. Jibladze, Compact Hausdorff spaces with relations and Gleason spaces, to appear in
*Appl. Categorical Structures 27 (2019), no. 6, 663-686 .* - J. Harding, Modularity is not canonical,
*Alg. Univ*.80 (2019), no. 1, Paper No. 8, 4 pp. - J. Harding, C. Heunen, B. Lindenhovious, and M. Navara, Boolean subalgebras of orthoalgebras,
*Order**Order*36 (2019), no. 3, 563-609. - G. Bezhanishvili, J. Harding, J. Ilin, and F. Lauridsen, MacNeille transferability and stable classes of Heyting algebras,
*Alg. Univ.*79 (2018), no. 3. - J. Harding, C. Walker and E. Walker, The convolution algebra,
*Alg. Univ.*79 (2018), no. 2. - J. Harding, Dynamics in the decompositions approach to quantum mechanics,
*Internat. J. of Theoret. Phys.*56, (2017), no. 12, 3971-3990. - J. Harding, Wigner’s theorem for an infinite set,
*Math. Slovaca*68 (2018), no. 5, 1173-1222. - J. Harding and A. Romanowska, Varieties of Birkhoff systems part I,
*Order*34 (2017) no. 1, 45-68. - J. Harding and A. Romanowska, Varieties of Birkhoff systems part II,
*Order*34 (2017) no. 1, 69-89. - J. Harding, C. Walker, and E. Walker, The Type-2 Truth Value Algebra, CRC Press, 2016.
- A. Doering and J. Harding, Abelian subalgebras and the Jordan structure of a von Neumann algebra,
*The Houston J. of Math.*42 (2016) no. 2, 559-568*.* - G. Bezhanishvili and J. Harding, On the proof that compact Hausdorff Boolean algebras are power sets,
*Order*33 (2016) no. 2, 263-268. - G. Bezhanishvili and J. Harding, Compact Hausdorff Heyting algebras,
*Algebra Universalis*76 (2016) 301-304. - J. Harding and Taewon Yang, Sections in orthomodular structures of decompositions,
*The Houston J. Math*. 42 (2016) no. 4, 1079-1092. - J. Harding and Tim Hannan, Automorphisms of Decompositions,
*Math. Slovaka*66 (2016) no. 2, 493-526. Sage code for programs - J. Harding and T. Yang, The logic of bundles,
*Internat. J. of Theoret. Physics*54 (2015) no. 12, 4601-4614*.* - J. Harding, C. Walker, and E. Walker, Partial orders on fuzzy truth value algebras,
*Internat. J. of Uncertainty, Fuzziness, and Knowledge-based Systems,*23 (2015), no. 2, 193-219. - G. Bezhanishvili, N. Bezhanishvili, and J. Harding, Modal operators on compact regular frames and de Vries algebras,
*Applied Categorical Structures,*23 (2015), no. 3, 365-379. - G. Bezhanishvili, N. Bezhanishvili, and J. Harding, Modal compact Hausdorff spaces,
*Journal of Logic and Computation,*25 (2015) no. 1, 1-35. - J. Harding, C. Walker, and E. Walker, Equations in type-2 fuzzy sets,
*Inter. J. of Uncertainty, Fuzziness, and Knowledge-Based Systems*23 (2015), 31-42. - G. Bezhanishvili and J. Harding, Stable Compactifications of frames,
*Cahiers de Topologie et Geometrie Differentielle Categoriques*55 (2014) no. 1, 37-65. - G. Bezhanishvili and J. Harding, Proximity frames and regularization,
*Applied Categorical Structures,*22 (2014), no. 1, 43-78. - J. Harding, C. Walker and E. Walker, Categories with fuzzy sets and relations,
*Fuzzy Sets and Systems,*256 (2014), no. 1, 43-78. - J. Harding, Daggers, kernels, Baer *-semigroups, and orthomodularity,
*J. Phil. Logic*42 (2013) no. 3, 535-549. - J. Harding, Decidability of the equational theory of the continuous geometry CG(F),
*J. Phil. Logic*42 (2013), no. 3, 461-465. - J. Harding, C. Walker, and E. Walker, Type II fuzzy sets and bichains, an invited chapter for Recent Advances in Type-2 Fuzzy Sets and Systems — Theory and Applications”, a book in the series
*Studies in Fuzziness and Soft Computing*vol. 301, pg. 97-112, 2013. - J. Harding, A Boolean topological orthomodular poset,
*Algebra Universalis*68 (2012), no. 3-4, 193-196. - J. Harding, C. Walker, and E. Walker, Projective bichains,
*Algebra Universalis*67 (2012), no. 4, 347-374. - G. Bezhanishvili and J. Harding, Modal logics of Stone spaces,
*Order*29 (2012), no. 2, 271-292. - J. Harding and Qin Yang, Regular completions of lattices.
*Houston J. of Mathematics*38 (2012), no. 3., 685-691. - J. Harding and M. Navara, Subalgebras of orthomodular lattices,
*Order*28 (2011), 549-563. - J. Harding, C. Walker, and E. Walker, The variety generated by the truth value algebra of type-II fuzzy sets,
*FuzzySets and Systems*161 (2010), no. 5, 735 – 749. - J. Harding, C. Walker, and E. Walker, Convex normal functions revisited,
*Fuzzy Sets and Systems*161 (2010), 1343 – 1349. - Q. Deng, J. Harding, and T. Hu, Hausdorff dimension of self-similar sets with overlaps,
*Science*in China, Series A: Mathematics 52 (2009), no. 1, 119 – 128. - J. Harding, A link between quantum logic and categorical quantum mechanics, International J. of Theoretical Physics (2009), no. 3, 769 – 802.
- G. Bezhanishvili and J. Harding, The modal logic of β(N), Archiv. Math. Logic. 48 (2009), 231 – 242.
- J. Harding, κ – complete uniquely complemented lattices, Order 25 (2008), no. 2, 121 – 129.
- J. Harding, Completions of Ordered Algebraic Structures: A Survey, invited chapter for the Proceedings of the International Workshop on Interval/Probabilistic Uncertainty and Non-classical Logics, Ono et. al. Ed.s, Advances in Soft Computing vol. 46, 2008, Springer, 231 – 244.
- J. Harding, C. Walker, and E. Walker, Lattices of convex normal functions, Fuzzy Sets and Systems 159 (2008), no. 9, 1061 – 1071.
- J. Harding, A regular completion for the variety generated by the three-element Heyting algebra, The Houston J. of Math. 34 (2008), no. 3, 649 – 660
*.* - J. Harding, The Source of the Orthomodular Law, a book chapter in The Handbook of Quantum Logic and Quantum Structures, Elsevier, 2007.
- Bezhanishvili and J. Harding, MacNeille completions of modal algebras, The Houston. J of Math. 33 (2007), no. 2, 355 – 384.
- J. Harding, Orthomodularity of decompositions in a categorical setting. International J. of Theoretical Physics 45 (2006), no. 6, 1117 – 1128.
- M. Gehrke, J. Harding, Y. Venema, MacNeille completions and canonical extensions. Trans. Amer. Math. Soc. 358 (2006), no. 2, 573 – 590.
- J. Harding, On profinite completions and canonical extensions, Algebra Universalis 55 (2006), no. 2-3, 293 – 296.
- J. Harding, D. Smith and E. Jager, Group-valued measures on the lattice of closed subspaces of a Hilbert space. International J. of Theoretical Physics. 44 (2005), no. 5, 539 – 548.
- G. Bezhanishvili, M. Gehrke, J. Harding, C. Walker and E. Walker, Varieties of Algebras that arise in Fuzzy Set Theory. Logical, algebraic, analytic, and probabilistic aspects of triangular norms, 321 – 344, Elsevier, Amsterdam, 2005.
- J. Harding, Remarks on concrete orthomodular lattices. International J. of Theoretical Physics 43 (2004), no. 10, 2149 – 2168.
- G. Bezhanishvili and J. Harding, MacNeille completions of Heyting algebras. The Houston J. of Math. 30 (2004), no. 4, 937 – 952.
- J. Harding and M. Roddy, Obituary: Günter Bruns. Order 20 (2004), pp. 329-332.
- J. Harding, The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators.
*Algebra Universalis*, 48 (2) (2002), pp. 171-182. - G. Bezhanishvili and J. Harding, Functional monadic Heyting algebras.
*Algebra Universalis*, 48 (1) (2002), pp. 1-10. - J. Harding and P. Ptak, On the set representation of an orthomodular poset.
*Coll. Math.*89 (2) (2001), pp. 233-240. - J. Harding, States on orthomodular posets of decompositions.
*International J. of Theoretical Physics*40 (2001), pp. 1061-1069. - M. Gehrke and J. Harding, Bounded lattice expansions.
*J. of Algebra*238 (2001), pp. 345-371. - G. Bruns and J. Harding, Algebraic aspects of orthomodular lattices,
*Current Research in Operational Quantum Logic: Algebras, Categories, Languages,*B. Cooke, D. Moore and A. Wilce ed., Kluwer 2000. - J. Harding and M. Navara, Embeddings into orthomodular lattices with given centers, state spaces and automorphism groups.
*Order*17 (2000), pp. 239-254. - G. Bruns and J. Harding, Epimorphisms in certain varieties of algebras.
*Order*17 (2000), pp. 195-206. - J. Harding and A. Pogel, Every lattice with 1 and 0 is embeddable in the lattice of topologies of some set by an embedding which preserves the 1 and 0.
*Topology and Its Applications*105 (2000), pp. 99-101. - J. Harding, The axioms of an experimental system.
*International J. of Theoretical Physics*38 (6) (1999), pp. 1643-1675. - J. Harding, Regularity in quantum logic.
*International J. of Theoretical Physics*37 (4) (1998), pp. 1173-1212. - J. Harding, Canonical completions of lattices and ortholattices.
*Tatra Mountains Math. Publ.*15 - G. Bruns and J. Harding, Amalgamation of ortholattices.
*Order*14 (1998), pp. 193-209. - J. Harding, M. Marinacci, N. Nguyen, and T. Wang, Local Radon-Nikodym derivatives of set functions.
*International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems*5 (3) (1997), pp. 379-394. - J. Harding and M. F. Janowitz, A bundle representation for continuous geometries.
*Advances in Applied Math.*19 (1997), pp. 282-293. - J. Harding, Decompositions in quantum logic.
*The Trans. Amer. Math. Soc.*348 (5) (1996), pp. 1839-1862. - G. D. Crown, J. Harding, and M. F. Janowitz, Boolean products of lattices.
*Order*13 (2) (1996), pp. 175-205. - J. Harding, Free central extensions.
*The Houston J. of Math.*22 (4) (1996), pp. 665-686. - J. Harding, The MacNeille completion of a uniquely complemented lattice.
*The Canad. Math. Bull.*37 (2) (1994), pp. 222-227. - J. Harding, Completions of orthomodular lattices II.
*Order*10 (1993), pp. 283-294. - J. Harding, Any lattice can be regularly embedded into the MacNeille completion of a distributive lattice.
*The Houston Journal of Math.*19 (1993), pp. 39-44. - J. Harding, Irreducible orthomodular lattices which are simple.
*Algebra Universalis*29 (1992), pp. 556-563. - J. Harding, Orthomodular lattices whose MacNeille completions are not orthomodular.
*Order*8 (1991), pp. 93-103. - J. Harding, Sheaves of orthomodular lattices and MacNeille completions. Ph.D. thesis. McMaster University, 1991.
- G. Bruns, R. J. Greechie, J. Harding, and M. Roddy, Completions of orthomodular lattices.
*Order 7*(1990), pp. 67-76. - J. Harding, Boolean factors of orthomodular lattices.
*Algebra Universalis*25 (1988), pp. 281-282. - J. Harding, Varieties of ortholattices containing the orthomodular lattices. M.Sc. thesis. McMaster University, 1988.