Free inverse semigroupoids and their inverse subsemigroupoids

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Semigroupoids are generalizations of semigroups and of small categories. In general, the quotient of a semigroupoid by a congruence is not a semigroupoid and homomorphisms of semigroupoids can also behave badly. We define certain types of congruences and homomorphisms that avoid this problem. We then investigate inverse semigroupoids which are semigroupoids in which each element has a unique inverse. A free inverse semigroupoid has a (symmetric) basis, and it turns out to be unique. Using the immersion of graphs from Stallings folding, we introduce the Stallings kernel. We use this to study the structure of free inverse semigroupoids and their inverse subsemigroupoids. We show that closed inverse subsemigroupoids of a free inverse semigroupoid are to some extent similar to subgroups of a free group. In particular, there are analogues of the Nielsen-Schreier theorem and Howson's theorem. In contrast to the situation in a free group, every finitely generated closed inverse subsemigroupoid of a free inverse semigroupoid F has finite index (whether or not F is finitely generated).

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