Abelian/commutative semigroups
Euclidian rings extend rings with operations for division and
remainder, with the Euclidian definition:
plus(mul(div(s, t), t), mod(s, t)) === s
,
with f(mod(s, t)) in [0, abs(t))
for some appropriate
embedding into real numbers.
Euclidian rings extend rings with operations for division and
remainder, with the Euclidian definition:
plus(mul(div(s, t), t), mod(s, t)) === s
,
with f(mod(s, t)) in [0, abs(t))
for some appropriate
embedding into real numbers.
Fields are commutative rings in which all non-zero elements have multiplicative inverses.
Groups are monoids that additionally have inverses
Monoids are semigroups with a neutral element (or zero)
Ordered rings are rings with ordering relation in which
addition, multiplication, and ordering are consistent:
leq(s, t) ==> leq(plus(s, a), plus(t, a))
and
leq(zero, s) & leq(zero, t) ==> leq(zero, mul(s, t))
.
Ordered rings are rings with ordering relation in which
addition, multiplication, and ordering are consistent:
leq(s, t) ==> leq(plus(s, a), plus(t, a))
and
leq(zero, s) & leq(zero, t) ==> leq(zero, mul(s, t))
.
A Pseudo-ring is a structure with the same operations as a ring, but no guarantee that multiplication satisfies the ring axioms
Rings are structures with both addition and multiplication
Rings that also have a division operation (though possibly not satisfying the standard axioms)
Ring that can also convert ring elements back to integers.
Rings that also possess an ordering relation
The additive group of integers
The built-in ring of integers
Package object making available some of the objects in sub-packages