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By Andrew O Lindstrum

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Again there are many structures that can be viewed as a collection of sets, and thus become a poset under this same containment relation: for example, subspaces of a vector space, subspaces of a point-line geometry, subgroups of a group, ideals in a ring and R-submodules of a given R-module and in general nearly any admissible subobject of an object admitting some specified set of algebraic properties. Two elements x and y are said to be incomparable if both of the statements x ≤ y and y ≤ x are false.

The real number system R. 5. Any induced subposet of a totally ordered set. We have already mentioned intervals of the real line. (Remark: the word “interval” here is used for the moment as it is used in Freshman College Algebra, open, closed, and half-open intervals such as (a, b] or [a, ∞). In this context, the intervals of posets that we have defined earlier, become the closed intervals, [a, b], of the real line, with a consistency of notation. Here is an example: Consider the induced poset of the rational numbers (Q, ≤) consisting of those positive fractions less than or equal to 1/2 which (in lowest terms) have a denominator not exceeding the positive integer d in absolute value.

Exception to Rule #2: For right R-modules we indicate the right operators by right “multiplication”, that is right juxtaposition. Ring multiplication still gets represented the right way since we have (mr )s = m(r s) for module element m and ring elements r and s (it looks like an associative law). 2 Binary Operations and Monoids It is not our intention to venture into various algebraic structures at such an early stage in this book. But we are forced to make an exception for monoids, since they are always lurking around so many of the most basic definitions (for example, the definition of interval measures on posets).

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