CAUL2012

WORKING SEMINARS ON NONCOMMUTATIVE LATTICES

by João Pita Costa

University of Ljubljana, Slovenia

Abstract: Pascual Jordan was the first to study noncommutative lattices in 1949. Skew lattices have been the most successful variation of noncommutative lattices. Jonathan Leech studied a more general version of these algebras and was later interested in their Boolean version termed skew Boolean algebras. The left-handed version of that case includes the class of Boolean skew algebras earlier studied by W.D. Cornish. R.J. Bignall, following ideas of Keimal and Werner, observed a subclass of skew Boolean algebras constitutes a decidable discriminator variety. In collaboration with J. Leech, R. Veroff, R.J. Bignall and M. Spinks have studied general properties of these algebras and used them in the study of multiple valued logic. A special attention has been always devoted to skew lattices in rings, that constitute a large class of examples, where Karin Cvetko-Vah and JPC answered several open questions. Today the classical dualities as Stone’s and Priestley’s are a focus of research in this context, where several relevant results have been achieved.

This workshop is distributed into three independent sessions. In the beginning of every session the preliminaries needed for it will be reviewed so that the availability of the visitor permits the frequency to most of the sessions. Each of them is also self-contained, presupposing only basic knowledge on Lattice Theory and Semigroup Theory. For further information please contact joaopitacosta ( at ) gmail.com.

Besides the available bibliography pointed for each session, most of the course notes and material we follow during this workshop is present on this author's Ph. D. dissertation. A wikipedia review article was developed by the author together with K. Cvetko-Vah and J. Leech which can be of interest. To get to know more about the recent advances on Skew Lattice Theory, please visit the project here or keep in touch with the news feed developed by Joao Pita Costa on the subject.

Several are the tools to work with these algebras, some of them comprehending computer reasoning. Prover9 is an automated theorem prover for first-order and equational logic, and Mace4 searches for finite models and counterexamples. A ready-to-use list of axioms was developed by the author (and K. Cvetko-Vah) during the preparation of his Ph. D. thesis.

#1 Skew Lattice Theory

4.10.2012, 15:00, FCUL - Room 6.2.44

Keywords: Basic concepts. Rectangular structure. Order structure. Decomposition theorems. Coset structure decomposition.

In this first session we are going to introduce the basic notions of the theory of skew lattices under the framework of the study of noncommutative lattices (cf. [12], [16], [28] and [29]). The importance of their nature as double regular bands of semigroups will permit us to import from semigroup theory to skew lattice theory two important decompositions from results due to Clifford, McLean and Kimura (cf. [6], [13], [18] and [24]). Furthermore, we will introduce the coset structure decomposition of a skew lattice, due to J. Leech (cf. [21]), introduce categorical skew lattices (cf. [14], [21] and [25]), and discuss its impact on the further study of the order structure of a skew lattice, and its combinatorial consequences.

Exercises Here. Please contact Joao Pita Costa at joaopitacosta (at) gmail for more details.

#2 Varieties of Skew Lattices

8.10.2012, 14:30, IIIUL - Room B3-01

Keywords: Distributivity and cancellation. Varieties of skew lattices. Skew lattices in rings.

Unlike what happens in lattice theory, the properties of cancellation and distributivity are independent in skew lattice theory (cf. [9]). The further study of these properties unveils several important varieties of skew lattices that have an important role and relationships of interest (cf. [8], [18], [20], [22] and [30]). Several of these varieties can be characterized by the description of their coset structure (cf. [11], [21], [25] and [26]). These are determined by properties like normality, symmetry, cancellativity or distributivity. We will also look at one of the most important class of examples, skew lattices in rings, that satisfies many of these studied properties (cf. [5], [7], [10] and [18]).

Exercises Here. Please contact Joao Pita Costa at joaopitacosta (at) gmail for more details.

#3 Skew Boolean Algebras

12.10.2012, 14:30, IIIUL - Room B3-01

Keywords: Skew Boolean algebras. Congruences. Subdirectly irreducibles. Aspects of Stone’s duality for skew lattices.

A Boolean version of these noncommutative lattices has been subject of interest since the early beginning of their study (cf. [4], [17], [18], [19] and [23]). A left-handed version of these algebras, under certain restrictions, proves to be congruence distributive constituting a discriminator variety (cf. [3] and [4]). The importance of skew Boolean lattices is shown in the behavior of their congruences. They also constitute a large class of examples by satisfying most of the studied properties. A generalization of Stone's duality for these algebras was recently achieved and will be a matter of discussion in this last session (cf. [1], [15] and [27]). We shall also refer to aspects of the generalization of Priestley duality for certain distributive skew lattices (cf. [2]).

Exercises Here. Please contact Joao Pita Costa at joaopitacosta (at) gmail for more details.

Bibliography:

[1] A. Bauer and K. Cvetko-Vah. Stone duality for skew boolean algebras with intersections. arXiv:1106.0425, Houston Journal of Mathematics (to appear, 2012)

[2] A. Bauer, K. Cvetko-Vah, M. Gehrke, S. van Gool, G. Kudryavtseva A non-commutative Priestley duality. arXiv:1206.5848 (2012).

[3] R. Bignall and J. Leech. Skew boolean algebras and discriminator varieties. Algebra Universalis, 33:387–398 (1995).

[4] W. H. Cornish. Boolean skew algebras. Acta Mathematica Academiae Scientiarurn Hungarkcae, Tomus 36(3-4):281–291 (1980).

[5] K. Cvetko-Vah. Skew lattices of matrices in rings. Algebra Universalis, 53:471–479 (2005).

[6] K. Cvetko-Vah. Internal decompositions of skew lattices. Communincations in Algebra, 35:243–247 (2007).

[7] K. Cvetko-Vah. On the structure of semigroups of idempotent matrices. Linear Algebra and its Applications, 429:204–213 (2007).

[8] K. Cvetko-Vah. On strongly symmetric skew lattices. Algebra Universalis, 66:99-113 (2011).

[9] Cvetko-Vah, K., Kinyon, M., Leech, J., Spinks, M.: Cancellation in skew lattices. Order, 28:9–32 (2011)

[10] K. Cvetko-Vah and J. Pita Costa. On the coset laws for skew lattices in rings. Novi Sad J. Math, 40(3):11–25, (2010).

[11] K. Cvetko-Vah and J. Pita Costa. On coset laws for skew lattices. Semigroup Forum, 83:395–4011 (2011)

[12] P. Jordan. Uber nichtkommutative verbande. Arch. Math, 2:56–59 (1949).

[13] N. Kimura. The structure of idempotent semigroups. Pacific J. Math., 8:257–275 (1958).

[14] M. Kinyon and J. Leech. Categorical skew lattices. arXiv:1201.3033 (2012).

[15] G. Kudryavtseva. A refinement of Stone duality to skew Boolean algebras. Algebra Universalis 67(4):397-416 (2012).

[16] G. Laslo and J. Leech. Green’s equivalences on noncommutative lattices. Acta Sci. Math., 68:501–533 (2002).

[17] J. Leech. Towards a theory of noncommutative lattices. Semigroup Forum, 34:117–120 (1986).

[18] J. Leech. Skew lattices in rings. Algebra Universalis, 26:48–72 (1989).

[19] J. Leech. Skew Boolean algebras. Algebra Universalis, 27:497–506 (1990).

[20] J. Leech. Normal skew lattices. Semigroup Forum, 44(1-8) (1992).

[21] J. Leech. The geometric structure of skew lattices. Trans. Amer. Math. Soc., 335:823– 842 (1993).

[22] J. Leech. Recent developments in the theory of skew lattices. Semigroup Forum, 52:7–24 (1996).

[23] J. Leech and M. Spinks. Skew boolean algebras derived from generalized boolean algebras. Algebra Universalis, 58:287–302 (2008).

[24] D. McLean. Idempotent semigroups. The American Mathematical Monthly, 61(2):110–113 (1954).

[25] J. Pita Costa. Coset laws for categorical skew lattices. Algebra Universalis. DOI: 10.1007/s00012-012-0194-z (2012).

[26] J. Pita Costa. On the coset structure of a skew lattice. Demonstratio Mathematica, 44(4):1–19 (2011).

[27] J. Pita Costa. On ideals for skew lattices. Discussiones Mathematicae, General Algebra and Applications (in press, 2012).

[28] V. Slav´ık. On skew lattices i. Comment. Math. Univer. Carolinae, 14:73–85 (1973).

[29] V. Slav´ık. On skew lattices ii. Comment. Math. Univer. Carolinae, 14:493–506 (1973).

[30] M. Spinks. Automated deduction in non-commutative lattice theory. Technical report, Gippsland School of Comp. and Inform. Tech., Monash Univ. (1998).

Acknowledgment: we would like to thank to K. Cvetko-Vah that was of great help in the preparation of this workshop, and to M. J. Gouveia for the invitation and the support to our work.

Location: Centro de Álgebra da Universidade de Lisboa

Instituto para a Investigação Interdisciplinar da Universidade de Lisboa

Av. Prof. Gama Pinto, 2

1649-003 Lisboa

Portugal