On ideals of a skew lattice

João Pita Costa

Discussiones Mathematicae - General Algebra and Applications, Tome 32 (2012), p. 5-21 / Harvested from The Polish Digital Mathematics Library

Résumé

Ideals are one of the main topics of interest when it comes to the study of the order structure of an algebra. Due to their nice properties, ideals have an important role both in lattice theory and semigroup theory. Two natural concepts of ideal can be derived, respectively, from the two concepts of order that arise in the context of skew lattices. The correspondence between the ideals of a skew lattice, derived from the preorder, and the ideals of its respective lattice image is clear. Though, skew ideals, derived from the partial order, seem to be closer to the specific nature of skew lattices. In this paper we review ideals in skew lattices and discuss the intersection of this with the study of the coset structure of a skew lattice.

Détail

Publié le : 2012-01-01

Zbl 1298.06002

EUDML-ID : urn:eudml:doc:270188

Bibliographie

[000] [1] A. Bauer and K. Cvetko-Vah, Stone duality for skew Boolean algebras with intersections, Arxiv preprint arXiv:1106.0425, Houston Journal of Mathematics (to appear, 2012). | Zbl 1284.06032

[001] [2] R. Bignall and J. Leech, Skew Boolean algebras and discriminator varieties, Algebra Universalis 33 (1995) 387-398. doi: 10.1007/BF01190707 | Zbl 0821.06013

[002] [3] G. Birkhoff, Lattice Theory, volume 5, AMS Colloquium Publications (Providence RI, third edition, 1940).

[003] [4] W.H. Cornish, Boolean skew algebras, Acta Mathematica Academiae Scientiarurn Hungarkcae 36 (3-4) (1980) 281-291. doi: 10.1007/BF01898144 | Zbl 0465.06010

[004] [5] K. Cvetko-Vah and J. Pita Costa, On the coset laws for skew lattices, Semigroup Forum 8 (2011) 395-411. doi: 10.1007/s00233-011-9325-7 | Zbl 1270.06003

[005] [6] G. Grätzer, Lattice Theory (San Francisco, WH Freeman and Co, 1971).

[006] [7] J.M. Howie, An Introduction to Semigroup Theory (Academic Press, 1976). | Zbl 0355.20056

[007] [8] P. Jordan, The mathematical theory of quasiorder, semigroups of idempotents and noncommutative lattices - a new field of modern algebra, Technical report, Armed Services Technical Information Agency (Arlington, Virginia, 1961).

[008] [9] M. Kinyon and J. Leech, Categorical Skew Lattices, arXiv:1201.3033 (2012).

[009] [10] J. Leech, Towards a theory of noncommutative lattices, Semigroup Forum 34 (1986) 117-120. doi: 10.1007/BF02573155 | Zbl 0607.06004

[010] [11] J. Leech, Skew lattices in rings, Algebra Universalis 26 (1989) 48-72. doi: 10.1007/BF01243872 | Zbl 0669.06006

[011] [12] J. Leech, Skew boolean algebras, Algebra Universalis 27 (1990) 497-506. doi: 10.1007/BF01188995 | Zbl 0719.06010

[012] [13] J. Leech, Normal skew lattices, Semigroup Forum 44 (1992) 1-8. doi: 10.1007/BF02574320

[013] [14] J. Leech, The geometric structure of skew lattices, Trans. Amer. Math. Soc. 335 (1993) 823-842. doi: 10.1090/S0002-9947-1993-1080169-X | Zbl 0792.06008

[014] [15] J. Leech, Recent developments in the theory of skew lattices, Semigroup Forum 52 (1996) 7-24. doi: 10.1007/BF02574077 | Zbl 0844.06003

[015] [16] J. Leech and M. Spinks, Skew Boolean algebras derived from generalized Boolean algebras, Algebra Universalis 58 (2008) 287-302. doi: 10.1007/s00012-008-2069-x | Zbl 1146.06008

[016] [17] J. Pita Costa, Coset Laws for Categorical Skew Lattices (Algebra Univers., in press, 2011).

[017] [18] J. Pita Costa, On the coset structure of skew lattices, Demonstratio Mathematica 44 (4) (2011) 1-19.

[018] [19] J. Pita Costa, On the Coset Structure of Skew Lattices (PhD thesis, University of Ljubljana, 2012).

[019] [20] B.M. Schein, Pseudosemilattices and pseudolattices, Amer. Math. Soc. Transl. 119 (1983) 1-16. | Zbl 0502.06001

[020] [21] V. Slavík, On skew lattices I, Comment. Math. Univer. Carolinae 14 (1973) 73-85. | Zbl 0261.06005

[021] [22] W. Wechler, Universal Algebra for Computer Scientists (Springer-Verlag, Berlin, 1992). doi: 10.1007/978-3-642-76771-5