Abstract: | In this talk, we introduce several classes of monoids satisfying up to five axioms and establish basic theories on their arithmetics. The one satisfying all the axioms is named natural monoid. Two typical examples are 1) the monoid of natural numbers in the group of positive rationals and 2) a certain monoid in one of Thompson's groups. The latter one is non-abelian, which serves as an important example for non-commutative arithmetics. In particular, defining primes in a non-abelian monoid is highly non-trivial, which relies on a concept we called "castling". |