# Introduction to universal al...

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**Speaker：**

**Prof.**

**Ralph McKenzie**

**（**

**Vanderbilt University**

**）**

**Title：**

**I**

**ntroduction 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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