On the semigroups of fence-decreasing and fence-preserving transformations on a finite fence

For a natural number n , a fence [ n ] = {1 ≺ 2 ≻ 3 ≺ 4 ≻ 5 ≺⋯ n } is a partial ordered set. A partial transformation α is called fence-decreasing if xα ⪯ x for all x in the domain of α , and fence-preserving if x ≺ y implies xα ⪯ yα for all x and y in the domain of α . In this paper, we consider the monoids D F n DFₙ ( P D F n ) (PDFₙ) of all fence-decreasing full (partial) transformations as well as the monoid P C F n PCFₙ of all fence-preserving transformations of P D F n PDFₙ . For these three monoids and some of their ideals, we determine the unique minimal generating set. Moreover, we calculate the rank of D F n DFₙ , P D F n PDFₙ , and P C F n PCFₙ . Additional, we provide several combinatorial results concerning these three monoids.