# On ideals of a skew lattice

Discussiones Mathematicae - General Algebra and Applications (2012)

- Volume: 32, Issue: 1, page 5-21
- ISSN: 1509-9415

## Access Full Article

top## Abstract

top## How to cite

topJoão Pita Costa. "On ideals of a skew lattice." Discussiones Mathematicae - General Algebra and Applications 32.1 (2012): 5-21. <http://eudml.org/doc/270188>.

@article{JoãoPitaCosta2012,

abstract = {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.},

author = {João Pita Costa},

journal = {Discussiones Mathematicae - General Algebra and Applications},

keywords = {noncommutative lattice; skew lattice; band of semigroups; ideals; coset structure; Green's relations; skew Boolean algebras},

language = {eng},

number = {1},

pages = {5-21},

title = {On ideals of a skew lattice},

url = {http://eudml.org/doc/270188},

volume = {32},

year = {2012},

}

TY - JOUR

AU - João Pita Costa

TI - On ideals of a skew lattice

JO - Discussiones Mathematicae - General Algebra and Applications

PY - 2012

VL - 32

IS - 1

SP - 5

EP - 21

AB - 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.

LA - eng

KW - noncommutative lattice; skew lattice; band of semigroups; ideals; coset structure; Green's relations; skew Boolean algebras

UR - http://eudml.org/doc/270188

ER -

## References

top- [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). Zbl1284.06032
- [2] R. Bignall and J. Leech, Skew Boolean algebras and discriminator varieties, Algebra Universalis 33 (1995) 387-398. doi: 10.1007/BF01190707 Zbl0821.06013
- [3] G. Birkhoff, Lattice Theory, volume 5, AMS Colloquium Publications (Providence RI, third edition, 1940).
- [4] W.H. Cornish, Boolean skew algebras, Acta Mathematica Academiae Scientiarurn Hungarkcae 36 (3-4) (1980) 281-291. doi: 10.1007/BF01898144 Zbl0465.06010
- [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 Zbl1270.06003
- [6] G. Grätzer, Lattice Theory (San Francisco, WH Freeman and Co, 1971).
- [7] J.M. Howie, An Introduction to Semigroup Theory (Academic Press, 1976). Zbl0355.20056
- [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).
- [9] M. Kinyon and J. Leech, Categorical Skew Lattices, arXiv:1201.3033 (2012).
- [10] J. Leech, Towards a theory of noncommutative lattices, Semigroup Forum 34 (1986) 117-120. doi: 10.1007/BF02573155 Zbl0607.06004
- [11] J. Leech, Skew lattices in rings, Algebra Universalis 26 (1989) 48-72. doi: 10.1007/BF01243872 Zbl0669.06006
- [12] J. Leech, Skew boolean algebras, Algebra Universalis 27 (1990) 497-506. doi: 10.1007/BF01188995 Zbl0719.06010
- [13] J. Leech, Normal skew lattices, Semigroup Forum 44 (1992) 1-8. doi: 10.1007/BF02574320
- [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 Zbl0792.06008
- [15] J. Leech, Recent developments in the theory of skew lattices, Semigroup Forum 52 (1996) 7-24. doi: 10.1007/BF02574077 Zbl0844.06003
- [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 Zbl1146.06008
- [17] J. Pita Costa, Coset Laws for Categorical Skew Lattices (Algebra Univers., in press, 2011).
- [18] J. Pita Costa, On the coset structure of skew lattices, Demonstratio Mathematica 44 (4) (2011) 1-19.
- [19] J. Pita Costa, On the Coset Structure of Skew Lattices (PhD thesis, University of Ljubljana, 2012).
- [20] B.M. Schein, Pseudosemilattices and pseudolattices, Amer. Math. Soc. Transl. 119 (1983) 1-16. Zbl0502.06001
- [21] V. Slavík, On skew lattices I, Comment. Math. Univer. Carolinae 14 (1973) 73-85. Zbl0261.06005
- [22] W. Wechler, Universal Algebra for Computer Scientists (Springer-Verlag, Berlin, 1992). doi: 10.1007/978-3-642-76771-5

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.