In the semigroup, we introduce a generalized definition of the inverse. A semigroup in which each element has exactly one inverse is called an inverse semigroup. Here, we define natural partial order and introduce the concept of congruence.
We mainly focus on the free inverse semigroup, which is defined as a certain quotient semigroup of free semigroup with involution. In the bachelor's thesis, we solve the word problem, i.e., we describe an algorithm that determines when two words represent the same element in a free inverse semigroup. There are several possible approaches to the solution. We present the solution using Scheiblich normal forms and we briefly describe Munn's algorithm.
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