Gillibert Pierre, Jonušas Julius, Pinsker Michael
Institut für Diskrete Mathematik und Geometrie Technische Universität Wien Wiedner Hauptstrasse 8-10/104 1040 Wien Austria.
Department of Algebra Charles University Prague Sokolovská 83 Prague 186 75 Prague 8 Czech Republic.
Bull Lond Math Soc. 2019 Oct;51(5):917-936. doi: 10.1112/blms.12286. Epub 2019 Sep 12.
About a decade ago, it was realised that the satisfaction of a given (or ) of the form in an algebra is equivalent to the algebra forcing a loop into any graph on which it acts and which contains a certain finite subgraph associated with the identity. Such identities have since also been called , and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a in any arbitrary non-trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form , which we call . We show that their satisfaction in an algebra is equivalent to any action of the algebra on a certain type of relation forcing a constant tuple into the relation. Proving that for each fixed width there is a weakest loop condition (that is, one entailed by all others), we obtain a new and short proof of the recent celebrated result stating that there exists a concrete loop condition of width 3 which is entailed in any non-trivial idempotent, possibly infinite, algebra. The framework of classical (width 2) loop conditions is insufficient for such proof. We then consider pseudo-loop conditions of finite width, a generalisation suitable for non-idempotent algebras; they are of the form , and of central importance for the structure of algebras associated with -categorical structures. We show that for the latter, satisfaction of a pseudo-loop condition is characterised by , that is, loops modulo the action of the automorphism group, and that a weakest pseudo-loop condition exists (for -categorical cores). This way we obtain a new and short proof of the theorem that the satisfaction of any non-trivial identities of height 1 in such algebras implies the satisfaction of a fixed single identity.
大约十年前，人们认识到，在一个代数中满足给定形式（或）的等式，等同于该代数将一个环强制引入到它所作用的任何包含与单位元相关的特定有限子图的图中。此后，这样的等式也被称为[具体名称未给出]，并且这种特征描述在泛代数中产生了惊人的结果，例如在任何非平凡有限幂等代数中满足[具体等式未给出]。从这个角度出发，我们开始系统地研究形如[具体形式未给出]的等式集，我们称之为[具体名称未给出]。我们证明，在一个代数中满足这些等式，等同于该代数对某种关系的任何作用将一个常数元组强制引入到该关系中。通过证明对于每个固定的宽度，存在一个最弱的环条件（即被所有其他条件所蕴含的条件），我们得到了一个新的简短证明，即最近著名的结果表明存在一个宽度为3的具体环条件，它在任何非平凡幂等（可能是无限的）代数中都成立。经典（宽度为2）环条件的框架不足以进行这样的证明。然后我们考虑有限宽度的伪环条件，这是一种适用于非幂等代数的推广；它们的形式为[具体形式未给出]，并且对于与[具体内容未给出] - 范畴结构相关的代数结构至关重要。我们表明，对于后者，伪环条件的满足由[具体内容未给出]来刻画，即模自同构群作用的环，并且存在一个最弱的伪环条件（对于[具体内容未给出] - 范畴核）。通过这种方式，我们得到了一个新的简短证明，即定理：在这样的代数中满足任何高度为1的非平凡等式意味着满足一个固定的单一等式。
