researchstatementskew

Research Statement

1. Introduction. While Algebra is a wide field of Mathematics that has proven to bring great contributions to other natural sciences like chemistry or biology (through group theory for instance), it is nowadays an important base of knowledge for the recent developments in Computer Science and other fields of knowledge. Universal Algebra studies algebraic structures that satisfy certain common properties and is able to determine general laws that hold in different fields of Algebra as Linear Algebra and Semigroup Theory. It is of great importance today hosting the Constraint Satisfaction Problem a subject of intense research of interest of both artificial intelligence and operations research, since the regularity in their formulation provides a common basis to analyze and solve problems of many unrelated families. This research by João Pita Costa aims the further development of Lattice Theory, that is one of the most important topics of study in Universal Algebra, in interdisciplinary connection with topics studied by Topology and Logic, formalized in what is known as Duality Theory. The pertinence of this research is due to the recent high quality results, by renown mathematicians as K. Cvetko-Vah, A. Bauer, G. Kudryavtseva and Mai Gehrke, the interest for the further research in this topic at the Algebra Center of the University of Lisbon and the possibility of intra-European international collaboration that is now arising to this topic of research where the applicant is an experienced researcher.

Universal algebra is the field of Mathematics that studies algebraic structures, constituted by a set and a collection of n-ary operations defined in it (including constants considered as nullary operations) ; and morphisms (maps preserving operations) between them. The nature of the algebra can be determined by axioms, which in universal algebra often take the form of identities, or equational laws (an example is the associative axiom for a binary operation). A variety is a collection of algebraic structures which satisfy by a common identities. These are sufficiently important that some authors consider varieties the only object of study in universal algebra. The term Universal Algebra remains unaltered since Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898. In the early 1930s, Garrett Birkhoff and Øystein Ore began publishing on this research topic. Although the development of mathematical logic had made applications to algebra possible, they came about slowly (results published by Anatoly Maltsev in the 1940s went unnoticed because of the war). In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoff's papers dealing with free algebras, congruence and subalgebra lattices, and homomorphism theorems. Lattices are of great importance in the study of Universal Algebra permitting the construction of clear examples and revealing its presence in many circumstances. These algebraic structures are determined by sets with two associative, idempotent and commutative binary operations that satisfy the property of absorption that establishes a relation between them. Developments in metamathematics and category theory in the 1940s and 1950s furthered the field, particularly the work of Abraham Robinson, Alfred Tarski, Andrzej Mostowski, and their students (Brainerd 1967).

Pascual Jordan was the first to study non-commutative lattices in 1949 [16]. Skew lattices have been the most successful variation of non-commutative lattices. Jonathan Leech studied these algebras and was interested on skew Boolean algebras, that generalize Boolean algebras [20]. The variety of skew Boolean algebras includes the class of algebras earlier studied by W. D. Cornish in [6]. Of particular interest are the skew Boolean intersection algebras where every pair of elements has a natural meet with respect to the natural partial order. Following ideas of Keimal and Werner, R.J. Bignall observed that the variety of skew Boolean intersection algebras constitutes a discriminator variety and thus is both congruence permutable and congruence distributive (cf. [4]). J. Leech collaborated with R. Veroff, R. .J. Bignall and M. Spinks to study general properties of these algebras and use them in the study of multiple valued logic. A special attention has been devoted to skew lattices in rings that constitute a large class of examples; in particular K. Cvetko-Vah answered several open questions related with these algebras, often in collaboration with J. Leech (cf. [9]). Unlike what happens in lattices, the properties of cancellation and distributivity are independent for skew lattices. K. Cvetko-Vah, J. Leech, M. Spinks and M. Kinyon studied both of these properties in [8] and [17], and unveiled several characterizations and subvarieties that enriched the theory at a great level and permitted the topic of research of João Pita Costa’s Ph.D. dissertation to contribute to an important and actual discussion. The work of these authors was presented at the Algebra Center of the University of Lisbon (CAUL) by M. Kinyon on the occasion of Workshop on Algebra and Logic in 2009. In the following years the applicant continued the dissemination of the new accomplishments in Skew Lattice Theory at CAUL, in the period between 2009 and 2012, while developing his Ph.D. research at the University of Ljubljana.

2. Aims & Objectives. The recent times have been very fruitful for the study of non-commutative lattices. Indeed, the increasing number of high quality mathematicians involved and in this topic of research, such as M. Kinyon or M. Gehrke, permits the engagement in achievements of great importance. Skew Boolean algebras have always been an essential class of examples of skew lattices that provides clear results that often derive from Lattice Theory and ensure the stability of the behavior of such algebraic structures. They consist of skew distributive lattices with bottom and well defined notion of relative complements. Recently, several important contributions have been made for the study of skew Boolean algebras such as the generalization of Stone’s duality for skew Boolean intersection algebras (cf. [28], [18], [19] and [1]). In fact, these algebras turn out to be dual to sheaves over locally-compact Boolean spaces. This duality can yield important new results and solve problems that have been opened earlier. This is the case of A. Bauer and K. Cvetko-Vah’s result answering negatively to the question of whether every skew lattice has a section, posed by J. Leech in the beginning of his research on the topic. With this research we shall accomplish the following listed objectives:

Systematic study of the coset structure of distributive skew lattices and its impact on the generalization of dualities in this context;

Exploration of some open problems on the recent generalization of Priestley’s duality for skew lattices using the new available approaches;

Construction of concrete canonical extensions of certain classes of skew lattices using topological representations;

Study of the dualizable varieties of skew lattices in the sense of the theory of natural dualities;

Contribution to the study of finitely generated varieties of skew lattices.

The relevance of such objectives is due to the contribution to the study of non-commutative lattices, within Lattice Theory and Universal Algebra, as well as to the general study of semigroups and its application to advanced Linear Algebra and Operator Theory - two of the main motivations for the study of such topic at the University of Ljubljana - as well as to Logic and Computer Science, with interesting new approaches being unveiled in the past ten years (cf. [8], [22], [3], [4], [5], [9] and [10]). The interdisciplinary status of such research will provide new approaches in these fields as it has proven to succeed to do in the recent past, through the works of J. Leech, R. Bignall or M. Spinks. The new machinery brought by the recently accomplished generalizations of Stone and Priestley dualities is of great importance and will ensure a new panoramic over open problems and research strategies (cf. [12], [13], [27] and [28]).

3. Significance. The ongoing research by K. Cvetko-Vah, A. Bauer, G. Kudryavtseva, Mai Gehrke and Sam Van Gool achieves a generalization of Priestley’s duality for distributive skew lattices, and opens the door to a great amount of relevant work to be done, with possible applications to computer science and operator theory (cf. [2] and [27]). Already in this setting it becomes apparent that the coset structure decomposition, studied by Pita Costa in his dissertation, plays an important role aiming the clear understanding of the range and applications of such duality. Many are the aspects that seem to be of coset nature on which Pita Costa can give a relevant contribution. Thus, the further research demands a systematic study of the coset structure of distributive skew lattices and its impact on the generalization of dualities in this context. The recently published scientific paper Coset laws for categorical skew lattices reveals innovative approaches to deal with this generalization of distributivity and is able to integrate a great deal of methods in this study with remarkable combinatorial consequences through the index theorems first presented in the mentioned scientific paper Cancellation in skew lattices published in Order in [8].

A further understanding of the generalization of important properties like distributivity and cancellation have opened many interesting problems in the literature. The recent works of M. Kinyon and J. Leech show the relevance of the coset structure approach to this study (cf. [21] and [17]). Moreover, this generalization of Priestley’s duality is of interest and relates to a stronger notion of distributivity, named skew distributivity (implying both distributivity and cancellation). Regarding the independence between distributivity and cancellation, it is relevant to consider such a generalization to both of these properties and the conditions on which this can be accomplished. When one will proceed towards such study, the cosets will be a crucial ingredient to build the dual étale space. The exploration of some open problems on the recent generalization of Priestley’s duality for skew lattices can thus profit of these new available approaches.

Moreover, the skew lattices are posets with two additional idempotent and associative binary operations satisfying the absorption laws. The notion of canonical extension of a lattice was already generalized to posets, while existence and uniqueness, up to isomorphism, were already proved for any poset. Hence we may consider canonical extensions of the poset reduct of skew lattices and lift up to it the binary operations of the skew lattices. We will build concrete canonical extensions of (particular) skew lattices using topological representations. There are varieties of skew lattices which are finitely generated as, for instance, the variety of skew quasi-Boolean algebras. The study of dualizability of such varieties within the theory of natural dualities is one of the topics of this research. For each algebra in such a variety its natural extension is isomorphic to its profinite limit and we will investigate how this algebra relates to the canonical extension of the algebra. With this we shall give a great contribution to the study of finitely generated varieties of skew lattices.

Furthermore, in every variety each algebra is a subdirect product of subdirecly irreducible algebras of the variety. The study of subdirectly irreducibles is of much relevance for the study of the varieties and its elements. We know that in the case of right-handed meet distributive skew lattices (or right-handed skew Boolean algebras) the only subdirectly irreducible algebras are 2 and 3_R. However, in the case of right-handed meet distributive skew lattices with intersections, all primitive skew lattices are subdirectly irreducible. We aim to get more information about the subdirectly irreducible skew lattices and consequently obtain more information about the different subvarieties of skew lattices accomplishing the description of relationships.

4. Methodology. The study of the variety of skew lattices stands in the intersection of Semigroup Theory and Lattice Theory, being consequently of great interest to both Universal Algebra and Linear Algebra, respectively. Much is known about lattices or bands of semigroups and several important results have been transferred to Skew Lattice Theory. Of great interest are the well known Leech’s Decomposition Theorems that describe the close relation between the variety of lattices and the variety of skew lattices, as well as the pullback that clarifies the relation of the flat with the full version of the algebra (cf. [22]). They permit the simplification of the study of general properties of skew lattices through the study of their left/right-handed versions.

Influenced by the study of bands of semigroups, we can define two quasiorders and an equivalence in any skew lattice, providing to it with a certain ordered structure. The equivalence coincides with Green’s relation D, which is a congruence in every skew lattice, and has an important role in the study of these algebras. The coset structure reveals the relationship between the congruence classes determined by the relation D. Every two related D-classes determine coset partitions in one another, enclosing interesting representation properties and unveiling the characterization of the partial order of the general algebra by isomorphisms between the correspondent cosets. This reveals a new perspective that does not have a counterpart either in the theory of lattices or in the theory of bands of semigroups (cf. [25]). The study of the coset structure of skew lattices began with Leech in [21], referred to as global coset geometry, and gives an insight into how the D-classes interact with each other in the lattice image of the general skew lattice thus providing important additional information.

This novel approach comprehends a perspective on skew lattices pointing out the decomposition of a skew lattice into D-classes and relating partitions that different D-classes induce on each other. This methods have been quite successful in the recent past. Of particular interest is the study of categorical skew lattices that has been in the center of attentions. The further development of the generalization of dualities in non-commutative lattices demands a systematic study of the coset structure of distributive skew lattices and a clarification of the impact of such approach on this topic of research. This issue was exposed recently by K. Cvetko-Vah at the Duality Theory workshop in Oxford University hosted by H. Priestley. The presence of G. Kudryavtseva as an invited speaker at the conference of Semigroups and Applications this year at the University of Uppsala was also relevant to the dissemination of the achieved new results that confirm that cosets will have a crucial role on the dual étale space, essential in the further research.

In his Ph.D. dissertation [24] Pita Costa, mentored by K. Cvetko-Vah, presented certain coset laws that describe the relation among the coset decompositions given by distinct pairs of D-classes in several varieties of skew lattices involved in the study of cancellative and distributive skew lattices (cf. [11], [23] and [25]). These coset laws permit the characterization of such varieties and the consequent establishment of relevant combinatorial results as mentioned in [8], the paper introducing the idea of index allowing us to count cosets. Pita Costa has given a series of talks on the subject at CAUL from 2009 to 2012, often at RAULA’s seminar, the project of CAUL under the responsibility of M. J. Gouveia. Between many other topics of research, Gouveia has collaborated with B. A. Davey and H. A. Priestley in the study of natural dualities and canonical extensions (cf. [12] and [14]). The research methodology and approach of natural dualities and of canonical extensions appropriate for the research field of non-commutative skew lattices contributing significantly to the development of the theory that, today, turns to the study of dualities, accomplishing results in matters of interest (cf. [1], [18] and [2]).

5. Bibliography.

[1] A. Bauer, K. Cvetko-Vah. Stone duality for skew Boolean algebras with intersections. Houston Journal of Mathematics (to appear).

[2] A. Bauer, K. Cvetko-Vah, M. Gehrke, S. Van Gool, G. Kudryavtseva. A non-commutative Priestley duality. arXiv: 1206.5848v1 (2012).

[3] J. Berendsen, D. Jansen, J. Schmaltz, and F.W. Vaandrager, The Axiomatization of Override and Update. Journal of Applied Logic 8 (2010), 141–150. DOI: 10.1016/j.jal.2009.11.001.

[4] R. Bignall, J. Leech. Skew Boolean algebras and discriminator varieties. Algebra Universalis 33 (1995), pp 387-398.

[5] R. Bignall and M. Spinks. Propositional skew boolean logic. In IEEE Comp. soc. Press, editor, 26th International Symposium on Multiple-Valued Logic (1996), pp 43–48.

[6] Carf, David, Cvetko-Vah, Karin. Skew lattice structures on the financial events plane. Appl. Sci. (2011), vol. 13, pp 9-20.

[7] W. H. Cornish. Boolean skew algebras. Acta Mathematica Academiae Scientiarum Hungarkcae 36 (1980), pp 281-291.

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

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

[10] K. Cvetko-Vah, J. Leech, M. Spinks. Skew lattices and binary operations on functions. (in preparation)

[11] K. Cvetko-Vah, J. Pita Costa. On the coset laws for skew lattices. Semigroup Forum 8 (2010), pp 395-411.

[12] B. A. Davey, M. J. Gouveia, M. Haviar, H. A. Priestley. Natural extensions and profinite completions of algebras. Algebra Universalis (to appear).

[13] M. Gehrke, H. Priestley. Canonical extensions and completions of posets and lattices. Reports on mathematical logic 43 (2008), pp 133-152.

[14] M. J. Gouveia. A note on profinite completions and canonical extensions. Algebra Universalis 64 (2010), pp 21-23.

[15] B. Davey, M. J. Gouveia, M. Haviar, H. A. Priestley. Natural extensions and profinite completions of algebras. Algebra Universalis (to appear).

[16] P. Jordan. Uber nichtkommutative verbande. Arch. Math. 2 (1949), pp 56-59.

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

[18] G. Kudryavtseva. A refinement of Stone duality for skew Boolean algebras. Alg. Universalis (2012), DOI 10.1007/s00012-012-0192-1.

[19] M. V. Lawson. A non-commutative generalization of Stone duality. J. Aus. Math Soc. 88 (2010), pp 385-404.

[20] J. Leech. Skew Boolean algebras. Algebra Universalis 27 (1990), pp 497-506.

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

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

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

[24] J. Pita Costa. On the coset structure of skew lattices. Ph. D. dissertation, University of Ljubljana (2012).

[25] J. Pita Costa. On the coset structure of a skew lattice. Demonstratio Mathematica Vol. XLIV No. 4. (2011), pp 673-692.

[26] J. Pita Costa. On ideals of a skew lattice. Discussiones Mathematicae (to appear).

[27] H. A. Priestley. Representation of distributive lattices by means of ordered stone spaces, Bull. London Math. Soc. 2 (1970), 186-190.

[28] M. H. Stone, The theory of representation for Boolean algebras. Trans. Am. Math. Soc. 74 (1936), 37-11

JOÃO PITA COSTA 2013