Abstract
The partial transformation semigroup [Formula: see text] is the semigroup of all partial transformations on the finite set n = {1,…, n}. The transformation semigroup [Formula: see text] and the symmetric group [Formula: see text] consist of all (full) transformations on n and permutations on n, respectively. We obtain presentations, in terms of generators and relations, for the singular subsemigroups [Formula: see text] and [Formula: see text]. We also calculate the ranks of both subsemigroups.
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