New perspectives on semi-primal varieties

Kurz, Alexander; POIGER, Wolfgang; TEHEUX, Bruno

doi:10.1016/j.jpaa.2023.107525

Download

Article (Scientific journals)

New perspectives on semi-primal varieties

Kurz, Alexander; POIGER, Wolfgang; TEHEUX, Bruno

2024 • In Journal of Pure and Applied Algebra, 228 (4), p. 107525

Peer Reviewed verified by ORBi

Permalink
https://hdl.handle.net/10993/57052

DOI
10.1016/j.jpaa.2023.107525

Files (1)Send to Details Statistics Bibliography Similar publications

Files

Full Text

KPT_SemiPrimalPaper.pdf

Author postprint (607.57 kB)

Download

All documents in ORBilu are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

Boolean power; Boolean skeleton; Canonical extension; Primal algebras; Semi-primal algebras; Stone duality; Algebra and Number Theory

Abstract :

[en] We study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.

Disciplines :

Mathematics

Author, co-author :

Kurz, Alexander; Chapman University, Orange, United States

POIGER, Wolfgang ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

TEHEUX, Bruno ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

yes

Language :

English

Title :

New perspectives on semi-primal varieties

Publication date :

April 2024

Journal title :

Journal of Pure and Applied Algebra

ISSN :

0022-4049

eISSN :

1873-1376

Publisher :

Elsevier B.V.

Volume :

228

Issue :

Pages :

107525

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0022404923002074?httpAccept=text/xml

Funders :

Fonds National de la Recherche Luxembourg

Funding text :

The second author is supported by the Luxembourg National Research Fund under the project PRIDE17/12246620/GPS .

Available on ORBilu :

since 04 October 2023

Statistics

Number of views

135 (8 by Unilu)

Number of downloads

107 (4 by Unilu)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

Adámek, J., Herrlich, H., Strecker, G.E., Abstract and Concrete Categories: The Joy of Cats. Reprints in Theory and Applications of Categories, vol. 17, 2006 Originally published by Wiley, New York, 1990.
Adámek, J., Milius, S., Sousa, L., Wissmann, T., On finitary functors. Theory Appl. Categ. 34:35 (2019), 1134–1164.
Baker, K.A., Pixley, A.F., Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z. 143 (1975), 165–174.
Banaschewski, B., Herrlich, H., Subcategories defined by implications. Houst. J. Math. 2 (1976), 149–171.
Barr, M., Fuzzy set theory and topos theory. Can. Math. Bull. 29:4 (1986), 501–508.
Bergman, C., Universal Algebra. Fundamentals and Selected Topics, 2011, Chapman and Hall/CRC.
Bergman, C., Bergman, J., Morita equivalence of almost-primal clones. J. Pure Appl. Algebra 108:2 (1996), 175–201.
Borceux, F., Handbook of Categorical Algebra, vol. 2. 1994, Cambridge University Press.
Burris, S., Boolean powers. Algebra Univers. 5 (1975), 341–360.
Burris, S., Discriminator varieties and symbolic computation. J. Symb. Comput. 13 (1992), 175–207.
Burris, S., Sankappanavar, H.P., A Course in Universal Algebra. Graduate Texts in Mathematics, vol. 78, 1981, Springer.
Chajda, I., Goldstern, M., Länger, H., A note on homomorphisms between products of algebras. Algebra Univers., 79, 2018.
Chang, C.C., Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88:2 (1958), 467–490.
Chang, C.C., A new proof of the completeness of the Łukasiewicz axioms. Trans. Am. Math. Soc. 93:1 (1959), 74–80.
Cignoli, R., Proper n-valued Łukasiewicz algebras as s-algebras of Łukasiewicz n-valued propositional calculi. Stud. Log. 41:1 (1982), 3–16.
Cignoli, R.L.O., D'Ottaviano, I.M.L., Mundici, D., Algebraic Foundations of Many-Valued Reasoning. Trends in Logic, vol. 7, 2000, Springer.
Clark, D.M., Davey, B.A., Natural Dualities for the Working Algebraist. Cambridge Studies in Advanced Mathematics, vol. 57, 1998, Cambridge University Press.
Cornish, W.H., Antimorphic Action: Categories of Algebraic Structures with Involutions or Anti-endomorphisms. Research and Exposition in Mathematics, vol. 12, 1986, Heldermann.
Davey, B.A., Gair, A., Restricted Priestley dualities and discriminator varieties. Stud. Log. 105 (2017), 843–872.
Davey, B.A., Haviar, M., Priestley, H.A., Bohr compactifications of algebras and structures. Appl. Categ. Struct. 25 (2017), 403–430.
Davey, B.A., Priestley, H.A., Canonical extensions and discrete dualities for finitely generated varieties of lattice-based algebras. Stud. Log. 100 (2012), 137–161.
Davey, B.A., Schumann, V., Werner, H., From the subalgebras of the square to the discriminator. Algebra Univers. 28 (1991), 500–519.
Esakia, L., Topological Kripke models. Sov. Math. Dokl. 15:1 (1974), 147–151.
Foster, A.L., Generalized “Boolean” theory of universal algebras. Part I. Math. Z. 58 (1953), 306–336.
Foster, A.L., Generalized “Boolean” theory of universal algebras. Part II. Identities and subdirect sums of functionally complete algebras. Math. Z. 59 (1953/1954), 191–199.
Foster, A.L., Semi-primal algebras; Characterization and normal-decomposition. Math. Z. 99 (1967), 105–116.
Foster, A.L., Automorphisms and functional completeness in universal algebras. Math. Ann. 180 (1969), 138–169.
Foster, A.L., Congruence relations and functional completeness in universal algebras. Structure theory of hemi-primals, I. Math. Z. 113 (1970), 293–308.
Foster, A.L., Pixley, A.F., Semi-categorical algebras. I. Semi-primal algebras. Math. Z. 83 (1964), 147–169.
Foster, A.L., Pixley, A.F., Semi-categorical algebras. II. Math. Z. 85 (1964), 169–184.
Galatos, N., Jipsen, P., Residuated lattices of size up to 6. Available online at https://math.chapman.edu/~jipsen/finitestructures/rlattices/RLlist3.pdf, 2017. (Accessed 4 September 2023)
Galatos, N., Jipsen, P., Kowalski, T., Ono, H., Residuated Lattices: An Algebraic Glimpse at Substructural Logics. 2007, Elsevier.
Gehrke, M., Jónsson, B., Bounded distributive lattice expansions. Math. Scand. 94:1 (2004), 13–45.
Goguen, J.A., L-fuzzy sets. J. Math. Anal. Appl. 18:1 (1967), 145–174.
Goguen, J.A., Concept representation in natural and artificial languages: axioms, extensions and applications for fuzzy sets. Int. J. Man-Mach. Stud. 6:5 (1974), 513–561.
Hu, T.-K., Stone duality for primal algebra theory. Math. Z. 110 (1969), 180–198.
Hu, T.-K., On the topological duality for primal algebra theory. Algebra Univers. 1 (1971), 152–154.
Hyland, M., Power, J., The category theoretic understanding of universal algebra: Lawvere theories and monads. Electron. Notes Theor. Comput. Sci. 172 (2007), 437–458.
Jipsen, P., Tsinakis, C., A survey of residuated lattices. Martínez, J., (eds.) Ordered Algebraic Structures: Proceedings of the Gainesville Conference Sponsored by the University of Florida, 28th February–3rd March, 2001, 2002, Springer US, Boston, 19–56.
Johnstone, P.T., Stone Spaces. 1982, Cambridge University Press.
Kaarli, K., Pixley, A.F., Polynomial Completeness in Algebraic Systems. 2001, Chapman and Hall/CRC.
Keimel, K., Werner, H., Stone duality for varieties generated by quasi-primal algebras. Mem. Am. Math. Soc. 148 (1974), 59–85.
Kowalski, T., Semisimplicity, EDPC and discriminator varieties of residuated lattices. Stud. Log. 77 (2004), 255–265.
Maruyama, Y., Natural duality, modality, and coalgebra. J. Pure Appl. Algebra 216:3 (2012), 565–580.
McKenzie, R., An algebraic version of categorical equivalence for varieties and more general algebraic categories. Ursini, A., Agliano, P., (eds.) Logic and Algebra Lecture Notes in Pure and Applied Mathematics, vol. 180, 1996, Routledge, New York, 211–243.
Moore, H.G., Yaqub, A., An existence theorem for semi-primal algebras. Ann. Sc. Norm. Super. Pisa 22:4 (1968), 559–570.
Moraschini, T., Raftery, J.G., Wannenburg, J.J., Varieties of De Morgan monoids: minimality and irreducible algebras. J. Pure Appl. Algebra 223:7 (2019), 2780–2803.
Murskiĭ, V.L., The existence of a finite basis of identities, and other properties of “almost all” finite algebras. Probl. Kibern. 30 (1975), 43–56.
Niederkorn, P., Natural dualities for varieties of MV-algebras I. J. Math. Anal. Appl. 255 (2001), 58–73.
nLab authors, Topological concrete category. Revision 38, available at https://ncatlab.org/nlab/revision/topological%20concrete%20category/38, 2022. (Accessed 4 September 2023)
Pixley, A.F., The ternary discriminator function in universal algebra. Math. Ann. 191 (1971), 167–180.
Porst, H.-E., Hu's primal algebra theorem revisited. Comment. Math. Univ. Carol. 41:4 (2000), 855–859.
Priestley, H.A., Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2:2 (1970), 186–190.
Quackenbush, R.W., Demi-semi-primal algebras and Mal'cev-type conditions. Math. Z. 122 (1971), 166–176.
Quackenbush, R.W., Primality: the influence of Boolean algebras in universal algebra. Grätzer, G., (eds.) Universal Algebra, second edition, 1979, Springer, New York, 401–416.
Ribes, L., Zalesskii, P., Profinite Groups. second edition, 2010, Springer.
Stone, M.H., The theory of representation for Boolean algebras. Trans. Am. Math. Soc. 40:1 (1936), 37–111.
Walker, C., Categories of fuzzy sets. Soft Comput. 8 (2004), 299–304.
Werner, H., Discriminator Algebras. Studien zur Algebra und ihre Anwendungen, vol. 6, 1978, Akademie Verlag.