Properties
The additive group of a partially ordered ring is always a partially ordered group.
The set of non-negative elements of a partially ordered ring (the set of elements x for which, also called the positive cone of the ring) is closed under addition and multiplication, i.e., if P is the set of non-negative elements of a partially ordered ring, then, and . Furthermore, .
The mapping of the compatible partial order on a ring A to the set of its non-negative elements is one-to-one; that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists.
If S is a subset of a ring A, and:
then the relation where iff defines a compatible partial order on A (ie. is a partially ordered ring).
In any l-ring, the absolute value of an element x can be defined to be, where denotes the maximal element. For any x and y,
holds.
Read more about this topic: Partially Ordered Ring
Famous quotes containing the word properties:
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—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)