MathDB

Let

\Gamma

be a variety of monoids such that not all monoids of

\Gamma

are groups. Prove that if

A \in \Gamma

and

B

is a submonoid of

A

, there exist monoids

S \in \Gamma

and

C

and epimorphisms

\varphi : S \rightarrow A, \;\varphi_1 : S \rightarrow C

such that ((e)\varphi_1^{\minus{}1})\varphi\equal{}B (

e

is the identity element of

C

). L. Marki

2