Explicit finite presentations -- generating sets plus complete lists of defining relations -- for the four monoids of graph maps of a finite star graph S_n: its endomorphisms, weak endomorphisms, strong endomorphisms, and strong weak endomorphisms. Establishes each monoid's rank (minimum generating-set size) and its regularity, and proves each presentation correct via the Guess-and-Prove canonical-form method, in which every word over the generators is rewritten to a unique normal form and the count of normal forms is matched to the monoid's order. Builds on the classical presentations of the full and partial transformation monoids and of the symmetric group.
A research paper in combinatorial semigroup theory (math.RA). For the star graph S_n (the tree on n+1 vertices consisting of a central vertex adjacent to n leaves), it studies four transformation mon…