Let In be the symmetric inverse semigroup, and let PODIn and POIn be its subsemigroups of monotone partial bijections and of isotone partial bijections on Xn = {1, . . . , n} under its natural order, respectively. | Turk J Math (2018) 42: 2270 – 2278 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Generating sets of certain finite subsemigroups of monotone partial bijections Leyla BUGAY∗,, Hayrullah AYIK, Department of Mathematics, Faculty of Science, Çukurova University, Adana, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: Let In be the symmetric inverse semigroup, and let P ODIn and P OIn be its subsemigroups of monotone partial bijections and of isotone partial bijections on Xn = {1, . . . , n} under its natural order, respectively. In this paper we characterize the structure of (minimal) generating sets of the subsemigroups P ODIn,r = {α ∈ P ODIn : |im (α)| ≤ r} , P OIn,r = {α ∈ P OIn : |im (α)| ≤ r} , and En,r = {id A ∈ In : A ⊆ Xn and |A| ≤ r} where idA is the identity map on A ⊆ Xn for 0 ≤ r ≤ n − 1 . Key words: Partial bijection, isotone/antitone/monotone map, (minimal) generating set 1. Introduction Let IX be the semigroup of all partial one-to-one maps on a nonempty set X under usual composition. It is well known that IX is an inverse semigroup; that is, for each element α there exists a unique element α′ such that αα′ α = α , which is called symmetric inverse semigroup. From the Wagner–Preston theorem, as cited in [4], as the analog of Cayley’s theorem for finite groups, every inverse semigroup is isomorphic to a subsemigroup of a suitable symmetric inverse semigroup. Hence, the symmetric inverse semigroups and their subsemigroups have certain important roles in inverse semigroup theory like the symmetric groups in group theory. Moreover, the problem of finding (minimal) generating sets of certain finite transformation semigroups is an important problem for finite semigroup theory and has been much studied over the last 50 years. We examine this problem for certain subsemigroups of In , the finite symmetric inverse semigroup on
