Detecting Modular ACU Structural Symmetries

Mapping Intimacies ◽

10.29007/mzj3 ◽

2018 ◽

Author(s):

María Alpuente ◽

Santiago Escobar ◽

Javier Espert

Keyword(s):

Automorphism Groups ◽

Directed Graphs

We present an efficient encoding of order-sorted modular ACU terms into colored directed graphs. Then, by computing the automorphism groups of the encoded graphs, we are able to extract ACU structural symmetries both inside a term and across a set of terms. Finally, we show how the computed symmetries can be applied to the optimization of the equational generalization algorithms for modular ACU theories.

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Amenability and unique ergodicity of automorphism groups of countable homogeneous directed graphs

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.84 ◽

2018 ◽

Vol 40 (5) ◽

pp. 1351-1401

Author(s):

MICHEAL PAWLIUK ◽

MIODRAG SOKIĆ

Keyword(s):

Automorphism Groups ◽

Directed Graphs ◽

Point Of View ◽

Unique Ergodicity ◽

Minimal Flow ◽

Negative Results ◽

Starting Point ◽

Complete Multipartite ◽

Uniquely Ergodic

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the ‘semi-generic’ complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of Angelet al[Random orderings and unique ergodicity of automorphism groups.J. Eur. Math. Soc.,16(2014), 2059–2095], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in Zucker [Amenability and unique ergodicity of automorphism groups of Fraïssé structures.Fund. Math.,226(2014), 41–61]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context of countable homogeneous directed graphs is given in Jasińskiet al[Ramsey precompact expansions of homogeneous directed graphs.Electron. J. Combin.,21(4), (2014), 31] and the papers cited therein.

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