set theory intersection is a multiplication (due to semigroup) #544 Correcting Math
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From: Archimedes Plutonium on 28 Mar 2010 16:25 Rotwang wrote: > > Part of the definition of a ring is having an additive inverse for each > element, which P(S) doesn't. It does form a "semiring" though. > > @A: another way in which intersection can be thought of as a product > (though I don't know of anybody using the term "multiplication" in this > context) is in the sense of category theory: the partial ordering on > P(S) makes it into a category, in which the intersection of any two sets > is a product (i.e. a greatest lower bound). Thanks, I looked it up and apparently it is considered multiplication in a semigroup, not semiring. --- quoting from http://www.cut-the-knot.org/do_you_know/mul_set.shtml --- Lattices have been introduced by the German mathematician J.W.R.Dedekind(1831-1916) along with his invention of ideals in rings. The word "lattice" was first circulated by the american G.D.Birkhoff (1884-1944) in 1930s. The definition is fantastically broad. In addition to set theory and ideals, numbers (integer and real) form a lattice if ab is defined as max(a,b) and the intersection of two numbers is set to be the minimum of the two. Now, returning to the product of two sets. As is well known, the frequently used notation for the intersection of two sets A nd B is plain AB. Regardless of the notations, it's a semigroup operation. --- end quoting from http://www.cut-the-knot.org/do_you_know/mul_set.shtml --- Archimedes Plutonium http://www.iw.net/~a_plutonium/ whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies |