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Erkko Lehtonen (Technische Universität Dresden): Graph algebras and graph varieties

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Wednesday, 26. September 2018, 10:00 - 12:00
Abstract. Graph algebras were introduced by Shallon in 1979. To each directed graph G=(V,E), we associate an algebra A(G) of type (2), whose universe is the set V∪{∞}, where ∞ is a new element not in V, and where the binary operation ("product") is defined as xy = x if (x,y)∈E and xy = ∞ otherwise. Encoding graphs as algebras in this way, we can view any algebraic properties of the graph algebra A(G) as properties of the graph G.

Although the class of graph algebras does not constitute a variety (as it is not closed under direct products), it makes perfect sense to consider the satisfaction relation between graphs (that is, graph algebras) and identities in the language of groupoids. Accordingly, the equational classes of graphs are called graph varieties. Graph varieties have been investigated by several authors. For example, Kiss, Pöschel and Pröhle determined the identities satified by all graphs. Poomsa-ard and his coauthors have characterized the graph varieties axiomatized by the transitive and the left or right self-distributive identities.

Continuing this line of research, we determined the graph varieties axiomatized by certain groupoid identities that are of general interest in algebra, such as the medial, (left or right) semimedial, idempotent, unipotent, zeropotent, and alternative identities.

This is joint work with Chaowat Manyuen (Khon Kaen University).
Location : Bolyai Intézet, I. emelet, Riesz terem, Aradi Vértanúk tere 1., Szeged


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