Department of Mathematics
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This pair of talks introduces ongoing research with Dr Anand Pillay of University of Notre Dame, Indiana.

In the first lecture I will define and discuss free algebras in a general setting. Then I will introduce the necessary elements of first order model theory to do the same for saturated models.

Freeness is a 'projective-like' property of an object \( A \): it implies the existence of morphisms with domain \( A \). Saturation is an 'injective-like' property of a structure \( A \): it implies the existence of morphisms with codomain \( A \). Taken together, these two properties place very strong restrictions on what the object \( A \) might look like. The canonical example is a vector space of sufficiently large dimension over a division ring.

I will go over the known results about saturated free algebras, and outline some of the new developments.

In the second lecture, I will continue by explaining what an indiscernible set is in model theory. One example of such is a free basis of a free algebra. The natural generalization of 'saturated free algebra' is 'saturated structure which is the algebraic closure of an indiscernible set'. I will go over known results about almost indiscernible theories, and discuss some of the new developments being considered by Pillay and myself.

This work is an extension and expansion of results of Pillay and Sklinos [2], which was itself a modernization and expansion of Baldwin and Shelah [1].

References:

1. Baldwin, J. T. and Shelah, S., The structure of saturated free algebras, Algebra Universalis 17(2):191–199 (1983)
2. Pillay, A. and Sklinos, R., Saturated free algebras revisited, Bull. Symb. Log., 21(3):306–318 (2015).