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
Summary
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.
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