Antiassociative groupoids

Abstract

Given a groupoid G, ⋆ , and k ≥ 3, we say that G is antiassociative iff for all x 1

Associative spectra of graph algebras I

“…It was also proved in [14] that there exist a continuum of different associative spectra (allowing infinite base sets, of course). Similar questions were investigated in [2][3][4], where some of the earlier results were rediscovered (with a different terminology).…”

“…Note the difference between (1) and ( 2): an arbitrary zag sequence can have arbitrarily large decreases, while a sequence satisfying (2) can drop at most by m − 2. 3 To prove the statement about Dyck paths, let us rewrite (2) in terms of the corresponding DFS tree T :…”

Term inequalities in finite algebras

“…This first application of f comes via the 3rd input, so we need to have f transfer the value in the 2nd component to an unused component, say the 1st. This gives us the linear transformation with equation f [1] := x 3 [2]. Since this transfers the value in the 2nd component to the 1st component along the path f 3 , our notation for this linear transformation will be 2, f 3 , 1 .…”

“…Thus s[0] is f (w, x, f (u, v, w)) [1]. To find the value of this, we use that the only piece of f assigning a value to the 1st component is 2, f 3 , 1 , and get f (w, x, f (u, v, w)) [1] = f (u, v, w) [2]. No piece of f assigns a value to the 2nd component, so f (u, v, w) [2] = 0, the default value.…”

“…Once we were looking at separating pairs of terms, it was natural to generalize this to groupoid terms s and t which had their variables appearing arbitrarily often in arbitrary orders. We solved the problem for many pairs of groupoid terms s and t in [2], showing that if s and t were ever separated in any groupoid, they were separated in a finite one.…”