Introduction to universal al...

2013-05-23 00:00 stxy View:[]

Speaker:Prof. Ralph McKenzieVanderbilt University
Title:Introduction to universal algebras
Schedule:

13, May, 9:20--10:20 am;

15, May, 10:00--11:00 am;

20, May, 9:20--10:20 am ;

22, May, 10:00--11:00 am;

27, May, 9:20--10:20 am;

29, May, 10:00--11:00 am.

Venue:Room 422 of School of Mathematics and Statistics
Content:
Lecture 1: Introducing algebras, both with and without signature, and their clone of operations and associated clone of relations. Homomorphisms, subalgebras and products.
Lecture 2: Congruences and homomorphism theorems. Exact polarity between operations and relations over finite sets.
Lecture 3: Free algebras, equations, varieties.
Lecture 4: Simple and subdirectly irreducible algebras in varieties. Elementary theory of semilattices, lattices and Boolean algebras
Lecture 5: Maltsev type conditions for permuting congruences, distributive (modular) congruence lattice, meet-semi-distributive congruence lattice.
Lecture 6: Abelian algebras, ternary groups, applications.
Lecture 7: Taylor-terms, cube-terms, Willard-terms
Lecture 8: A variety is congruence $SD(\wedge)$ iff it has no non-trivial Abelian congruences.
Lecture 9: Directly representable varieties.
References:
The book by Clifford Bergman [1] will be our primary source. The book [2] is quite good on most elementary topics. The book [3] covers these topics at a more advanced level. The monograph [5], and [4] on which it relies, present a new research direction which dominated universal algebra in the late twentieth century. The final four referenced papers are a sampling of the literature on the universal-algebraic approach to constraint satisfaction problems in theoretical computer science---a dominant theme in universal algebraic research for the past eight years.
[1] Clifford Bergman, Universal Algebra: Fundamentals and Selected Topics, CRC Press (a Chapman and Hall book) (2012).
[2] S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, Graduate Texts in Mathematics no. 78 (1981). Available for download at www.math.uwaterloo.ca/ snburris.)
[3] R. McKenzie, G. McNulty, W. Taylor, Algebras, Lattices, Varieties (I), Wadsworth and Brooks-Cole (1987). (Now available through Springer-Verlag.)
[4] D. Hobby, R. McKenzie, The Structure of Finite Algebras, American Mathematical Society, Contemporary Mathematics series no. 76 (1991).
[5] K. Kearnes, E. Kiss, The Shape of Congruence Lattices, American Mathematical Society, Memoirs no 1046 (2012).
[6] J. Berman, P. Idziak, P. Markov\'{i}c, R. McKenzie, M. Valeriote, R. Willard, Varieties with few subalgebras of powers, Transactions AMS 362 (2009), 1145--1173.
[7] L. Barto, M. Kozik and I. Niven, The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang-Jensen and Hell), SIAM Journal on Computing 38/5 (2009), pp. 1782--1802.
[8] L. Barto and M. Kozik, Absorbing subalgebras, cyclic terms, and the constraint satisfaction
problem, Logical Methods in Computer Science, vol. 8 (1:07) (2012), pp. 1--26.
[9] L. Barto, The dichotomy for conservative constraint satisfaction problems revisited, Proceedings of the 26th IEEE Symposium on Logic in Computer Science, LICS'11, pp. 301--310.

Brief:

Education:

Ph.D., University of Colorado, 1966

Professional Positions:

Professor, UC Berkeley, 1978–1994

Professor Emeritus, UC Berkeley, 1994–

Distinguished Professor, Vanderbilt University, 1994–

Awards and Honors:

Fulbright Senior Scholar, [Fall] 1989

Sloan Foundation Fellow, 1973–75

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