English

Loop conditions with strongly connected graphs

Logic 2018-10-09 v1 Combinatorics

Abstract

We prove that the existence of a term ss satisfying s(r,a,r,e)=s(a,r,e,a)s(r,a,r,e) = s(a,r,e,a) in a general algebraic structure is equivalent to an existence of a term tt satisfying t(x,x,y,y,z,z)=t(y,z,z,x,x,y)t(x,x,y,y,z,z)=t(y,z,z,x,x,y). As a consequence of a general version of this theorem and previous results we get that each strongly connected digraph of algebraic length one, which is compatible with an operation tt satisfying an identity of the from t()=t()t(\ldots)=t(\ldots), has a loop.

Cite

@article{arxiv.1810.03177,
  title  = {Loop conditions with strongly connected graphs},
  author = {Miroslav Olšák},
  journal= {arXiv preprint arXiv:1810.03177},
  year   = {2018}
}
R2 v1 2026-06-23T04:31:12.577Z