Polyadic analog of Grothendieck group

TL;DR

This paper extends the Grothendieck construction from binary to polyadic algebraic structures, revealing complex new properties and relationships between initial semigroups and resulting class groups.

Contribution

It introduces a generalized polyadic framework for Grothendieck groups, allowing different arities and noncommutative structures, expanding algebraic $K$-theory.

Findings

Polyadic completion groups have more complex structures than binary ones.

Different polyadic direct products can be constructed from a single semigroup.

The arity of class groups can differ from that of initial semigroups.

Abstract

We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.