# Bioperational Multisets in Various Semi-rings

## TL;DR

This paper explores bioperational multisets, which are sets with equal sum and product, across various algebraic domains including semi-rings, extending beyond the traditional whole numbers.

## Contribution

It systematically enumerates all sum-product pairs of bioperational multisets over whole numbers and six other algebraic domains, a novel extension in the study of such sets.

## Key findings

- Enumerated all sum-product pairs over whole numbers.
- Extended enumeration to six additional algebraic domains.
- Provided foundational data for further algebraic exploration.

## Abstract

One can find lists of whole numbers having equal sum and product. We call such a creature a bioperational multiset. No one seems to have seriously studied them in areas outside whole numbers such as the rationals, Gaussian integers, or semi-rings. We enumerate all possible sum-products for a bioperational multiset over whole numbers and six additional domains.