Originally published in an important series of books on pure and applied mathematics, this monograph by a distinguished mathematician explores a high-level area in algebra. It constitutes the first systematic summary of research concerning partially ordered groups, semigroups, rings, and fields. The self-contained treatment features numerous problems, complete proofs, a detailed bibliography, and indexes. It presumes some knowledge of abstract algebra, providing necessary background and references where appropriate. This inexpensive edition of a hard-to-find systematic survey will fill a gap in many individual and institutional libraries.
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Yes, you can access Partially Ordered Algebraic Systems by Laszlo Fuchs in PDF and/or ePUB format, as well as other popular books in Mathematics & Algebra. We have over one million books available in our catalogue for you to explore.
It will be convenient to collect here the fundamental concepts and terminology we shall need.
If a binary relation
is defined on a set A with the properties
then A is called a partially ordered set (abbreviated: p. o. set) and
is called a partial order on A. The dual of A is the p. o. set A′ with the same elements and with the partial order
′ defined as follows: a
′ b (in A′) if, and only if, b
a (in A).1
As usual, one may write b
a for a
b, and a < b (or b > a) to mean that a
b and a ≠ b. If neither a
b nor b
a, then a and b are called incomparable, in sign: a | | b.
It may happen that a relation
satisfies only P1 and P3; in this case we say
is a preorder.2 A preorder induces an equivalence relation ~ on A, namely, a ~ b if, and only if, simultaneously a
b and b
a. The set of classes a*, c*, … of this equivalence can be partially ordered in the natural way : a*
c* if for some (and hence for all) a in a* and for some (and so for all) c in c* we have a
c. The set A* of the classes a*, c*, ... is a p. o. set.
A partial order on A induces in the natural way a partial order on any non-void subset B of A) namely, for a, b ∈ B one puts a
b in B if, and only if, a
b in the original partial order of A. This induced partial order of B will be denoted by the same symbol
.
A (closed) interval3 [a, b] of A (where a
b) consists of all c ∈ A satisfying a
c
b; a and b are called the endpoints of [a, b], The subsets Ia = [x ∈ A | x
a], defined for each a ∈ A, and their duals Ja = [x ∈ A | x
a] are also considered as (closed) intervals. A subset of A is called convex if it contains the whole interval [a, b] whenever it contains the endpoints a, b. If we replace
by < we obtain the definition of open intervals (a, b), . . ..
Let A and A′ be p. o. sets. A mapping a → a′ from A into A′ is called isotone if it is single-valued and order-preserving in the sense that a
b implies a′
b′. A mapping which is one-to-one and isotone in both directions is said to be an isomorphism (or order-isomorphism) of A onto A′; A and A′ are then called isomorphic (order-iso...
