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Keywords:
equivalence $\tilde{\mathcal Q}^U$; left $C$-$\mathcal U$-liberal semigroup; left semi-spined product; band-formal construction; left $C$-liberal semigroup
Summary:
In this paper the equivalence $\tilde{\mathcal Q}^U$ on a semigroup $S$ in terms of a set $U$ of idempotents in $S$ is defined. A semigroup $S$ is called a $\mathcal U$-liberal semigroup with $U$ as the set of projections and denoted by $S(U)$ if every $\tilde{\mathcal Q}^U$-class in it contains an element in $U$. A class of $\mathcal U$-liberal semigroups is characterized and some special cases are considered.
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