Pseudo-loop conditions

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10401471" target="_blank" >RIV/00216208:11320/19:10401471 - isvavai.cz</a>

Result on the web

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=dixoNBz6DC" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=dixoNBz6DC</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1112/blms.12286" target="_blank" >10.1112/blms.12286</a>

Result language

angličtina

Original language name

Pseudo-loop conditions

Original language description

About a decade ago, it was realised that the satisfaction of a given identity (or equation) of the form f(x1, horizontal ellipsis ,xn)approximate to f(y1, horizontal ellipsis ,yn) 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 loop conditions, and this characterisation has produced spectacular results in universal algebra, such as the satisfaction of a Siggers identity s(x,y,z,x)approximate to s(y,x,y,z) in any arbitrary non-trivial finite idempotent algebra. We initiate, from this viewpoint, the systematic study of sets of identities of the form f(x1,1, horizontal ellipsis ,x1,n)approximate to MIDLINE HORIZONTAL ELLIPSIS approximate to f(xm,1, horizontal ellipsis ,xm,n), which we call loop conditions of width m. 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 m 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 u1 circle f(x1,1, horizontal ellipsis ,x1,n)approximate to MIDLINE HORIZONTAL ELLIPSIS approximate to um circle f(xm,1, horizontal ellipsis ,xm,n), and of central importance for the structure of algebras associated with omega-categorical structures. We show that for the latter, satisfaction of a pseudo-loop condition is characterised by pseudo-loops, that is, loops modulo the action of the automorphism group, and that a weakest pseudo-loop condition exists (for omega-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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2019

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Bulletin of the London Mathematical Society

ISSN

0024-6093

e-ISSN

—

Volume of the periodical

51

Issue of the periodical within the volume

5

Country of publishing house

GB - UNITED KINGDOM

Number of pages

20

Pages from-to

917-936

UT code for WoS article

000488368700013

EID of the result in the Scopus database

2-s2.0-85074199607