Abstract

FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra if its monoid of endomorphisms has commuting idempotents. We show that these conditions are not sufficient, by giving an example of an algebra satisfying the four properties, such that its monoid of endomorphisms does not have commuting idempotents. This settles a problem presented by FitzGerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible. Finally, we list some categories in which FitzGerald’s question has an affirmative answer.

Highlights

  • In universal algebra, an important invariant of an algebra is its monoid of endomorphisms

  • FitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra if its monoid of endomorphisms has commuting idempotents

  • We show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible

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Summary

Introduction

An important invariant of an algebra is its monoid of endomorphisms. Every variety of algebras has a full embedding in the category of directed graphs, see [6] It was shown by Hedrlín and Kucera that more generally any concrete category can be fully embedded in a variety of algebras, when working in von Neumann–Bernays– Gödel set theory (including the axiom of global choice), under the additional axiom that there are no measurable cardinals, see [4]. This motivates the following definition: a category is called (algebraically) universal if it contains a full subcategory equivalent to the category of directed graphs.

Background and statement of the problem
Counterexample to the problem
More general algebras
Categories for which FitzGerald’s problem has an affirmative answer
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