Post completely described clones on a 2-element set; there are only countably many. Yanov and Muchnik proved that already on 3 elements, there are uncountably many clones. These results may also be viewed as results about 2-element and 3-element structures considered up to primitive positive interdefinability. Motivated by research in constraint satisfaction, a coarser equivalence relation on finite structures has been introduced recently, based on primitive positive (pp) constructions instead of primitive positive definitions. On the algebraic side, this amounts to studying idempotent linear strong Maltsev conditions for clones on finite sets. We do not know how many finite structures that are up to pp interconstructability; all uncountable families of structures on finite domains known to the speaker collapse down to countably many with respect to pp interconstructability. I will present some recent results about the special case of finite directed graphs, where at least we know the largest finite digraphs with respect to pp constructability.
Joint work with Florian Starke, Albert Vucaj, and Dmitriy Zhuk
Finite structures ordered by pp constructability
Apr. 29, 2021 1pm (Zoom (vir…
Rep Theory
Gaywalee Yamskulna (Illinois State University)
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In this talk, I will discuss an impact of Leibniz algebras on the algebraic structure of -graded vertex algebras. Along the way, I will provide easy ways to characterize several types of -graded vertex algebras.