Talk:Reflexive Euclidean Relation is Equivalence

Isn't it true that if a binary relation is both left and right Euclidean, it is then an equivalence relation (without the need to specify that it is reflexive)? If aRb is true, then aRb and aRb is true, and left Euclidean implies then that aRa is true and right Euclidean implies then that bRb is true. PAR (talk) 13:35, 23 June 2015 (UTC)

No. For, who guarantees that for every $x$ there even exists a $y$ with $xRy$? — Lord_Farin (talk) 13:51, 23 June 2015 (UTC)

Ah, ok, thank you. I was implicitly assuming that was true without realizing it. I guess the correct statement would be that any total relationship which is left and right Euclidean is an equivalence relationship, where "total" means for every $x$ there exists a $y$ such that either $xRy$ or $yRx$ or both. PAR (talk) 14:26, 23 June 2015 (UTC)

Worth adding a new theorem to that effect: Total Euclidean Relation is Equivalence. --prime mover (talk) 11:43, 30 October 2021 (UTC)