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The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner–Preston Theorem. Let n be a positive natural number and PFI n be the semigroup of all fence-preserving partial one-to-one maps of { 1 , ... , n } into itself with respect to composition of maps and the fence 1 ≺ 2 ≻ 3 ≺ ⋯   n. There is considered the inverse semigroup IOF n par of all α ∈ PFI n such that α is regular in PFI n , order-preserving with respect to the order 1 < 2 < ⋯ < n and parity-preserving. According to the main result of the paper, it is 3 n − 6 the least of the cardinalities of the generating sets of IOF n par for 4 ≤ n. There is determined a concrete representation of a generating set of minimal size. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the study of congruences on semigroups of transformations on a countably infinite fence. We consider the monoid O ℕ of all full transformations on the set ℕ of all natural numbers preserving the zig-zag order on ℕ , as well as the monoid E (O ℕ ∗) of all idempotent transformations in O ℕ additionally preserving the usual linear order on ℕ. We show that there are uncountably many congruences on O ℕ and determine seven maximal congruences on O ℕ , which are all the maximal congruences containing a particular congruence on O ℕ. Moreover, we characterize all congruences on the monoid of all transformations in E (O ℕ ∗) with infinite rank. For the semigroup of all transformations in E (O ℕ ∗) with finite rank, we determine the Rees congruences. [ABSTRACT FROM AUTHOR]

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup FN of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that FN has no minimal generating sets and present two particular infinite decreasing chains of generating sets of FN. [ABSTRACT FROM AUTHOR]