Varieties of Regular Pseudocomplemented de Morgan Algebras
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Varieties of Regular Pseudocomplemented de Morgan Algebras ´ Vaz de Carvalho2 M. E. Adams1 · H. P. Sankappanavar1 · Julia Received: 24 July 2019 / Accepted: 28 November 2019 / © Springer Nature B.V. 2020
Abstract In this paper, we investigate the varieties Mn and Kn of regular pseudocomplemented de Morgan and Kleene algebras of range n, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in Mn and explicitly describe the dual spaces of the simple algebras in M1 and K1 . We show that the variety M1 is locally finite, but this property does not extend to Mn or even Kn for n ≥ 2. We also show that the lattice of subvarieties of K1 is an ω + 1 chain and the cardinality of the lattice of subvarieties of either K2 or M1 is 2ω . A description of the lattice of subvarieties of M1 is given. Keywords Variety of algebras · Lattice of subvarieties · Regular pseudocomplented de Morgan algebra (of range n) · Discriminator variety · Simple algebra · Subdirectly irreducible algebra · Priestley duality Mathematics Subject Classification (2010) Primary: 06D30, 06D15, 03G25 · Secondary: 08B15, 06D50, 03G10
1 Introduction A pseudocomplemented de Morgan algebra (pm-algebra for short) is an algebra (L; ∧, ∨,∗ , , 0, 1) of signature (2, 2, 1, 1, 0, 0) such that (L; ∧, ∨,∗ , 0, 1) is a pseudocomplemented distributive algebra and (L; ∧, ∨, , 0, 1) is a de Morgan algebra, that is H. P. Sankappanavar
[email protected] M. E. Adams [email protected] J´ulia Vaz de Carvalho [email protected] 1
Department of Mathematics, State University of New York, New Paltz, NY 12561, USA
2
Centro de Matem´atica e Aplicac¸o˜ es, Departamento de Matem´atica, Faculdade de Ciˆencias e Tecnologia, Universidade NOVA de Lisboa, 2829-516 Caparica, Portugal
Order
(L; ∧, ∨, 0, 1) is a bounded distributive lattice, ∗ satisfies, for x, y ∈ L, x ∧ y = 0 if and only if y ≤ x ∗ , whilst satisfies (x ∨ y) = x ∧ y , (x ∧ y) = x ∨ y , 0 = 1, 1 = 0, and x = x. Since, as shown by Ribenboim [19], the class of pseudocomplemented distributive algebras form a variety, so too does the class of pseudocomplemented de Morgan algebras. Pseudocomplemented de Morgan algebras were first investigated in Romanowska [20] and have been considered by several authors since (see, for example, Denecke [7], Gait´an [8], Guzm´an and Squier [9], X. Wang and L. Wang [28], as well as [21–24]). Romanowska’s characterisation of finite subdirectly irreducible pm-algebras in [20] was extended to a characterisation of all subdirectly irreducible pm-algebras in [21] (for those algebras which are non-regular) and in [23] (for those algebras which are regular). An algebra is regular if any two congruences on it are equal whenever they have a class in common. It was in [21] that regularity in pm-algebras was first considered. It is better understood in the context of double p-algebras. Regular double p-algebras have been studied extensively (se
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