Amalgamation Property in the subvarieties of Gautama and Almost Gautama Algebras

Document Type : Research Paper

Authors

1 Departamento de Matem´atica Universidad Nacional del Sur Alem 1253, Bah´ıa Blanca, Argentina INMABB - CONICET

2 Department of Mathematics State University of New York New Paltz, New York, 12561 U.S.A.

Abstract

Gautama algebras were introduced in 2022, as a common generalization of regular double Stone algebras and regular Kleene Stone algebras. Even more recently, Gautama algebras were further generalized to Almost Gautama algebras (AG for short). The main purpose of this paper is to investigate the Amalgamation Property (AP, for short) in the subvarieties of the variety AG. In fact, we show that, of the eight nontrivial subvarieties of AG, only four varieties, namely those of Boolean algebras, of regular double Stone algebras, of regular Kleene Stone algebras and of De Morgan Boolean algebras have the AP and the remaining four do not have the AP. We give several applications of this result; in particular, we examine the following properties for the subvarieties of AG: transferability property (TP), having enough injectives
(EI), Embedding Property, Bounded Obstruction Property and having a model companion. 

Keywords

Main Subjects

References

[1] Adams, M. E., Sankappanavar H. P., & Vaz de Carvalho, J. (2019). Regular double p-algebras, Mathematica Slovaca 69 (1), 15{34.
[2] Adams, M. E., Sankappanavar H. P., & Vaz de Carvalho, J. (2020). Varieties of regular pseudocomplemented De Morgan algebras, Order, 37(3), 529-557. https://doi.org/10.1007/s11083-019-09518-y.
[3] Albert, H., & Burris, S. (1988). Bounded obstructions, model companions and amalgamation bases, Math Logic Quarterly 34, 109-115.
[4] Balbes, R., & Dwinger, P. (1974). Distributive Lattices, Missouri Press.
[5] Bacsich, P. D. (1972). lnjectivity in model theory, Colloq. Math. 25, 165-176.
[6] Bacsich, P. D. (1972). Primality and model completions, Algebra Universalis 3, 265-270.
[7] Banaschewski, B. (1970). lnjectivity and essential extensions in equational classes of algebras, Proc. Conf. Universal Algebra (Queen's Univ., Kingston, Ont., October 1969) Queen's Papers in Pure and Applied Math. 25, Queen's Univ., Kingston, Ontario, 131-147.
[8] Bialynicki-Birula, A., & Rasiowa, H. (1957). On the representation of quasi-Boolean algebras. Bull. Acad. Polon. Sci. Cl. III 5:259-261, XXII.
[9] Bergman, C. (1983). The amalgamation class of a discriminator variety is  nitely axiomatizable. In: Freese, R.S., Garcia, O.C. (eds) Universal Algebra and Lattice Theory. Lecture Notes in Mathematics, vol 1004. Springer, Berlin, Heidelberg.
https://doi.org/10.1007/BFb0063427
[10] Bergman, C. (1983). Amalgamations classes of some distributive varieties, Algebra Universalis 20, 143-166.
[11] Boole, G. (1847). The Mathematical Analysis of Logic. Being an Essay towards a Calculus of Deductive Reasoning. Macmillan, Barclay, Macmillan, Cambridge.
[12] Boole, G. (1854). An Investigation of The Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. Originally published by Macmillan, London, 1854. Reprint by Dover, 1958.
[13] Bruyns P., Naturman C., Rose H.(1992). Amalgamation in the pentagon variety. Algebra Universalis 29, 303-322.
[14] Burris, S., & Sankappanavar, H. P. (1981). A course in universal algebra, Graduate Texts in Mathematics 78, Springer-Verlag, New York. The Millennium version (2012) is freely available online as a PDF  le at http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.
[15] Chang, C. C., & Keisler, H. J. (1973). Model theory, North-Holland, Amsterdam.
[16] Comer, S. D. (1969). Classes without the amalgamation property, Paci c J. Math. 28, No. 2.
[17] Cornejo, J. M., & Sankappanavar, H. P. (2022). A Logic for dually hemimorphic semi-Heyting algebras and its axiomatic extensions. Bulletin of the Section of Logic 51/4,555-645. https://doi.org/10.18778/0138-0680.2022.2391.
[18] Cornejo, J. M., Kinyon, M., & Sankappanavar, H. P. (2023). Regular double p-algebras: A converse to a Katrinaak theorem, and applications, Mathematica Slovaca. vol. 73, no. 6, 2023, 1373-1388. https://doi.org/10.1515/ms-2023-0099
[19] Cornejo, J. M., & Sankappanavar, H. P. (2023). Gautama and Almost Gautama algebras and their associated logics, Transactions on Fuzzy Sets and Systems (TFSS) 2(2), 77-112.
[20] Cornejo, J. M., & Sankappanavar, H. P. (2024). Quasi-Gautama algebras: A generalization of Almost Gautama algebras. Preprint.
[21] Cornejo, J. M., & Sankappanavar, H. P. (2024). A note on the equivalence of the Stone identity with Weak-star regularity in Almost Gautama algebras. Preprint.
[22] Czelakowski, J., & Pigozzi, D. (1996). Amalgamation and algebraic logic, Centre de Recerca Matematica, Bellaterra, preprint no. 343.
[23] Czelakowski, J., & Pigozzi, D. (1999). Amalgamation and interpolation in abstract algebraic logic. In: Models, Algebras, and Proofs selected papers of the X Latin American symposium on mathematical logic held in Bogota, edited by Xavier Caicedo and Carlos H. Montenegro, Lecture Notes in Pure and Applied Mathematics, Vol. 203, Marcel Dekker, Inc., New York.
[24] Frasse, R. (1954). Sur l'extension aux relations de quelques properietes des ordres, Ann. Sci. Ec. Norm. Super 71, 363-388.
[25] Gil-Ferez, J., Ledda, A., & Tsinakis, C. (2015). The Failure of amalgamation property for semilinear varieties of residuated lattices, Mathematica Slovaca 65, 817-828.
[26] Gratzer, G., and E.T. Schmidt. (1957). On a problem of M. H. Stone. Acta Mathematica Academiae Scientiarum Hungaricae 8: 455-460.
[27] Gratzer, G. (1971). LatticeTheory: First concepts and distributive lattices. Freeman, San Francisco.
[28] Gratzer, G., & Lakser, H. (1971). The structure of pseudocomplemented distributive lattices II: Congruence extension and amalgamation, Trans. Amer. Math. Soc. 156,, 343{358.
[29] Gratzer, G., Jonsson, B., & Lakser, H. (1973). The amalgamation property in equational classes of modular lattices, Paci c J. Math. 45, 507-524.
[30] Hirschfeld, J., Wheeler, W.H. (1975). Model-completions and model-companions. In: Forcing, Arithmetic, Division Rings. Lecture Notes in Mathematics, vol 454. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064085
[31] Jacobs, D. F. (1995). Amalgamation in varieties of algebras, Master's thesis, The university of Cape Town.
[32] Jenei, S. (2021). Amalgamation in classes of involutive commutative residuated lattices, arXiv:2012.14181v7 [math.LO] 14 Nov 2021.
[33] Jevons, W. S. (1864). Pure Logic, or the Logic of Quality apart from Quantity: with Remarks on Boole's System and on the Relation of Logic and Mathematics. Edward Stanford, Lon- don, 1864. Reprinted 1971 in Pure Logic and Other Minor Works ed. by R. Adamson and H.A. Jevons, Lennox Hill Pub. & Dist. Co., NY.
[34] Jonsson, B. (1956). Universal relational systems, Math. Scand. 4, 193-208.
[35] Jonsson, B. (1965). Extensions of relational structures, Proc. Internat. Sympos. Theory of Models (Berkeley, 1963), North-Holland, Amsterdam, 146-157.
[36] Jonsson, B. (1984). Amalgamation of pure embeddings, Algebra Universalis 19, 266-268.
[37] Jonsson, B. (1990). Amalgamation in small varieties of lattices, Journal of Pure and Applied Algebra 68, 195-208.
[38] Kalman, J. A. (1958). Lattices with involution. Trans. Amer. Math. Soc. 87, 485-491.
[39] Katrinak, J. A. (1973). The structure of distributive double p-algebras. Regularity and congruences, Algebra Universalis 3, 238{246,
[40] Kiiss, E. W., Marki, L., Prohle, P., & Tholen, W. (1983). Categorical algebraic properties. A Compendium on amalgamation, congruence extension, epimorphisms, residual smallness, and injectivity, Studia Scientiarum Mathematicarum Hungarica 18, 79-141.
[41] Koubek, V., Sichler, J. (1994). Amalgamation in varieties of distributive double palgebras, Algebra Universalis, 32, 407-438.
[42] Lipparini, P. (1982). Locally  nite theories with model companion. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 6-11.
[43] Madarasz, J. X. (1998). Interpolation and Amalgamation; Pushing the Limits. Part I, Studia Logica 61, 311-345.
[44] Metcalfe, G., Montagna, F., Tsinakis, C. (2014). Amalgamation and interpolation in ordered algebras, Journal of Algebra, 402, 21-82.
[45] McCune, W. (2011). Prover9 and Mace4, version 2009-11A, (http://www.cs.unm.edu/mccune/prover9/).
[46] Moisil, G. C. (1935). Recherches sur lalgebre de la logique, Annales scienti ques de luniversite de Jassy 22, 1-117.
[47] Moisil, G. C. (1942). Logique modale, Disquisitiones Mathematicae et Physica, 2, 3-98. Reproduced in pp. 341-431 of [48].
[48] Moisil, G. C. (1972). Essais sur les logiques non chrysippiennes, Editions de lAcademie de la Republique Socialiste de Roumanie, Bucharest,
[49] Monteiro, A. A. (1980). Sur les algebres de Heyting symetriques, Portugal. Math, 39(1-4), 1-237.
[50] Neumann, H. (1948). Generalized free products with amalgamated subgroups, Am. J. Math. 70, 590-625.
[51] Neumann, H. (1949). Generalized free products with amalgamated subgroups, Am. J. Math. 71, 491-540.
[52] Pierce, K. R. (1972). Amalgamations of lattice-ordered groups, Transactions of the American Mathematical Society 172, 249-260.
[53] Pierce, K. R. (1972). Amalgamating abelian ordered groups, Paci c Journal of Mathematics, 43 (3), 711-723.
[54] Pigozzi, D. (1971). Amalgamation, congruence-extension, and interpolation properties in algebras. Algebra Univ. 1, 269{349. https://doi.org/10.1007/BF02944991.
[55] Powell, W. B., & Tsinakis, C. (1982). Amalgamations of lattice-ordered groups. In: Ordered Algebraic Structures (W. B. Powell, C. Tsinakis, eds.). Lecture Notes in Pure and Appl. Algebra, Vol. 99, Marcel Dekker, New York-Basel, 171-178.
[56] Rasiowa, H. (1974). An algebraic approach to non-classical logics, Studies in Logic and the Foundations of Mathematics, Vol. 78, North-Holland Publishing Co., Amsterdam.
[57] Sankappanavar, H. P. (1986). Pseudocomplemented Okham and De Morgan algebras, Math. Logic Quarterly 32, 385{394.
[58] Sankappanavar, H. P. (1987). Semi-De Morgan algebras, J. Symbolic. Logic, 52, 712{724.
[59] Sankappanavar, H. P. (2011). Expansions of semi-Heyting algebras I: Discriminator varieties, Studia Logica, 98, no.1-2, 27{81.
[60] Sankappanavar, H. P. (2022). (Chapter) A Few Historical Glimpses into the Interplay between Algebra and Logic, and Investigations into Gautama Algebras. In: Handbook of Logical Thought in India, S. Sarukkai, M. K. Chakraborty (eds.), Springer Nature India Limited.
[61] Schreier, O. (1927). Die untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hambur 5, 161-183.
[62] Varlet, J. (1972). A regular variety of type (2, 2, I, 1, 0, 0), Algebra Universalis 2, 218-223.
[63] Werner, H. (1978). Discriminator algebras, Studien zur Algebra und ihre Anwendungen, Band 6, Academie-Verlag, Berlin.
[64] Yasuhara, M. (1974). The amalgamation property, the universal-homogeneous models and the generic models, Math. Stand. 34, 5-6. 
Journal of Mahani Mathematical Research
Volume 15, Issue 1 - Serial Number 33
January 2026
Pages 461-483

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  • Receive Date: 27 September 2025
  • Revise Date: 02 October 2025
  • Accept Date: 12 October 2025

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