Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T23:06:28.970Z Has data issue: false hasContentIssue false
  • English
  • Français

VARIETIES OF SKEW BOOLEAN ALGEBRAS WITH INTERSECTIONS

Part of: Varieties

Published online by Cambridge University Press:  09 September 2016

JONATHAN LEECH  and
MATTHEW SPINKS
JONATHAN LEECH
Affiliation:
Department of Mathematics, Westmont College, Santa Barbara, CA 93018, USA email leech@westmont.edu
MATTHEW SPINKS*
Affiliation:
Department of Philosophy, University of Cagliari, Cagliari 09123, Italy email mspinksau@yahoo.com.au
*
✉mspinksau@yahoo.com.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Skew Boolean algebras for which pairs of elements have natural meets, called intersections, are studied from a universal algebraic perspective. Their lattice of varieties is described and shown to coincide with the lattice of quasi-varieties. Some connections of relevance to arbitrary skew Boolean algebras are also established.

Keywords

skew Boolean algebrasintersections(sub)directly irreducible algebraslattice of (quasi-)varietiesdiscriminator varieties

MSC classification

Primary: 08B15: Lattices of varieties
Secondary: 06E75: Generalizations of Boolean algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 2 , April 2017 , pp. 290 - 306
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Bauer, A. and Cvetko-Vah, K., ‘Stone duality for skew Boolean intersection algebras’, Houston J. Math. 39 (2013), 73–109.Google Scholar
Bauer, A., Cvetko-Vah, K., Gehrke, M., van Gool, S. and Kudryavtseva, G., ‘A noncommutative Priestley duality’, Topology Appl. 160 (2013), 1423–1438.Google Scholar
Bignall, R. J., ‘Quasiprimal varieties, and components of universal algebras’, PhD Thesis, The Flinders University of South Australia, 1976.Google Scholar
Bignall, R. J. and Leech, J., ‘Skew Boolean algebras and discriminator varieties’, Algebra Universalis 33 (1995), 387–398.Google Scholar
Bignall, R. J. and Spinks, M., ‘Implicative BCS-algebra subreducts of skew Boolean algebras’, Sci. Math. Jpn. 58 (2003), 629–638.Google Scholar
Bignall, R. J. and Spinks, M., ‘Corrigendum: Implicative BCS-algebra subreducts of skew Boolean algebras’, Sci. Math. Jpn. 66 (2007), 387–390.Google Scholar
Blanco, J., Campercholi, M. and Vaggione, D., ‘The subquasivariety lattice of a discriminator variety’, Adv. Math. 159 (2001), 18–50.CrossRefGoogle Scholar
Blok, W. J. and Ferreirim, I. M. A., ‘On the structure of hoops’, Algebra Universalis 43 (2000), 233–257.Google Scholar
Blok, W. J. and Raftery, J. G., ‘Assertionally equivalent quasivarieties’, Int. J. Algebra Comput. 18 (2008), 589–681.Google Scholar
Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra, Graduate Texts in Mathematics, 78 (Springer, New York, 1981).Google Scholar
Campercholi, M., Stronkowski, M. and Vaggione, D., ‘On structural completeness versus almost structural completeness problem: a discriminator varieties case study’, Log. J. IGPL 23 (2015), 235–246.CrossRefGoogle Scholar
Cornish, W. H., ‘Boolean skew algebras’, Acta Math. Acad. Sci. Hungar. 36 (1980), 281–291.Google Scholar
Cvetko-Vah, K., ‘A new proof of Spinks’ theorem’, Semigroup Forum 73 (2006), 267–272.Google Scholar
Cvetko-Vah, K., ‘Internal decompositions of skew lattices’, Comm. Algebra 35 (2007), 243–247.Google Scholar
Cvetko-Vah, K. and Leech, J., ‘On maximal idempotent-closed subrings of M n (f)’, Comm. Algebra 21 (2011), 1097–1110.Google Scholar
Cvetko-Vah, K. and Leech, J., ‘Rings whose idempotents are multiplicatively closed’, Comm. Algebra 40 (2012), 3288–3307.Google Scholar
Cvetko-Vah, K., Leech, J. and Spinks, M., ‘Skew lattices and binary operations on functions’, J. Appl. Log. 11 (2013), 253–265.Google Scholar
Davey, B., ‘On the lattice of subvarieties’, Houston J. Math. 5 (1979), 183–192.Google Scholar
Kudryavtseva, G., ‘A refinement of Stone duality to skew Boolean algebras’, Algebra Universalis 67 (2012), 397–416.CrossRefGoogle Scholar
Kudryavtseva, G., ‘A dualizing object approach to noncommutative Stone duality’, J. Aust. Math. Soc. Ser. A 95 (2013), 383–403.Google Scholar
Kudryavtseva, G. and Leech, J., ‘Free skew Boolean algebras’, Int. J. Algebra Comput., to appear.Google Scholar
Leech, J., ‘Skew Boolean algebras’, Algebra Universalis 27 (1990), 497–506.CrossRefGoogle Scholar
Leech, J., ‘Normal skew lattices’, Semigroup Forum 44 (1992), 1–8.Google Scholar
Leech, J., ‘Recent developments in the theory of skew lattices’, Semigroup Forum 52 (1996), 7–24.Google Scholar
Leech, J. and Spinks, M., ‘Skew Boolean algebras derived from generalized Boolean algebras’, Algebra Universalis 58 (2008), 287–302.Google Scholar
Olson, J., Raftery, J. G. and van Alten, C. J., ‘Structural completeness in substructural logics’, Log. J. IGPL 16 (2008), 453–495.Google Scholar
Raftery, J. G., ‘Representable idempotent commutative residuated lattices’, Trans. Amer. Math. Soc. 359 (2007), 4405–4427.Google Scholar
Spinks, M., ‘On middle distributivity for skew lattices’, Semigroup Forum 61 (2000), 341–345.Google Scholar
Spinks, M., ‘Contributions to the theory of pre-BCK-algebras’, PhD Thesis, Monash University, 2003.Google Scholar
Spinks, M., Bignall, R. J. and Veroff, R., ‘Discriminator logics (research announcement)’, Australas. J. Log. 11 (2014), 159–171.Google Scholar
Veroff, R. and Spinks, M., ‘Axiomatizing the skew Boolean propositional calculus’, J. Automat. Reason. 37 (2006), 3–20.Google Scholar