A totally ordered monoid, or tomonoid for short, is a monoid endowed with a compatible total order. We reconsider in this paper the problem of describing the one-element Rees coextensions of a finite, negative tomonoid S, that is, those tomonoids that are by one element larger than S and whose Rees quotient by the poideal consisting of the two smallest elements is isomorphic to S. We show that any such coextension is a quotient of a pomonoid R(S) , called the free one-element Rees coextension of S. We investigate the structure of R(S) and describe the relevant congruences. We moreover introduce a finite family of finite quotients of R(S) from which the coextensions arise in a particularly simple way.
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