Is there an accepted name for algebraic structures like $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$?

Question

Question. Is there an accepted name for algebraic structures that, like $\mathbb{Q}_{>0}$ and $\mathbb{R}_{>0}$, are models of the algebraic theory presented as follows?

Sorts: $U$

Functions: $$+ : U \times U \rightarrow U$$ $$1 : U, \qquad \times : U \times U \rightarrow U, \qquad x \mapsto x^{-1} : U \rightarrow U$$

Axioms:

• $+$ is commutative and associative,

• The multiplicative structure forms an abelian group

• Multiplication distributes over addition.

Examples.

1. $\mathbb{Q}_{>0}$ (which is the initial such algebra)
2. $\mathbb{R}_{>0}$
3. $\mathbb{Z},$ with the symbol $\times$ interpreted as $+_\mathbb{Z}$, the symbol $1$ interpreted as the element $0_\mathbb{Z}$, the notation $x \mapsto x^{-1}$ interpreted as $x \mapsto -x$, and the symbol $+$ interpreted as either max or min.

Answer 1

That's a semfield. An additive identity might be additionally required.