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Euclidean semirings
Given a additively cancellative commutative semiring , a term is left cancellative if for all and , implies .
A term is right cancellative if for all and , implies .
An term is cancellative if it is both left cancellative and right cancellative.
The multiplicative submonoid of cancellative elements in is the subset of all cancellative elements in
A Euclidean semiring is a additively cancellative commutative semiring for which there exists a function from the multiplicative submonoid of cancellative elements in to the natural numbers, often called a degree function, a function called the division function, and a function called the remainder function, such that for all and , and either or .
Non-cancellative and non-invertible elements
Given a ring , an element is non-cancellative if: if there is an element with injection such that , then . An element is non-invertible if: if there is an element with injection such that , then .
An Archimedean ordered field is Cauchy if every Cauchy sequence of rational numbers in the field converges. The initial Cauchy field is the Cauchy real numbers.
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