Regular element of a monoid

For a ring $𝑅$ , an element $𝑥 \in 𝑅$ being regular is a positively defined way of saying $𝑥$ is not a zero-divisor. Thus it is particularly advantageous when LEM is not available. Another advantage is that being regular makes sense for arbitrary monoids.

Let $𝑀$ be a monoid. An element $𝑎 \in 𝑀$ is

left-regular iff $𝑎 𝑥 = 𝑎 𝑦 ⟹ 𝑥 = 𝑦$ or any $𝑥, 𝑦 \in 𝑀$ , i.e. $𝑎$ is monic in the delooping category $𝖡 𝑀$ ;
right-regular iff $𝑥 𝑎 = 𝑦 𝑎 ⟹ 𝑥 = 𝑦$ for any $𝑥, 𝑦 \in 𝑀$ , i.e. $𝑎$ is epic in the delooping category $𝖡 𝑀$ ;
regular iff it is left- and right-regular.

Assuming LEM one can show that an element of a ring is regular iff it is not a zero-divisor.¹

#state/tidy | #lang/en | #SemBr

See e.g. 2009. Algebra: Chapter 0, §III.1.2, ¶1.9, p. 122 ↩

Regular element of a monoid

Footnotes