From: Albert Lunde <Albert-Lunde@nwu.edu>

Date: Wed, 30 Aug 1995 11:35:26 -0600

Message-Id: <v01510100ac6a50785cbc@[129.105.110.129]>

To: http-wg%cuckoo.hpl.hp.com@hplb.hpl.hp.com

Date: Wed, 30 Aug 1995 11:35:26 -0600

Message-Id: <v01510100ac6a50785cbc@[129.105.110.129]>

To: http-wg%cuckoo.hpl.hp.com@hplb.hpl.hp.com

>>Idempotent > >I second the request for a clear definition in the context of this spec. >Another good reason for this: idempotent is not in any dictionary >(including my Webster's unabridged) that I could find. It's a mathematical term. Here's what our on-line OED says (special symbols are a bit mangled): >idempotent > >idempotent <e>ide;mpotent, <e>i:dempou.tent, , a. and sb. f. L. idem same + >potent-em >powerful, potent > >A. adj. Of a quantity or element a: having the property that a x a = a, >where x represents >multiplication or some other (specified) binary operation. Also applied to >an operator or set for which >this is true for any element a and to statements expressing this fact. > > 1870 B. Peirce in Amer. Jrnl. Math. (1881) IV. 104 When an >expression..raised to a square > or higher power..gives itself as the result, it may be called idempotent. > > 1937 A. A. Albert Mod. Higher Algebra (1938) iii. 88 A matrix E is >called idempotent if E2 > = E. > > 1937 Duke Math. Jrnl. III. 629 We recall that A &211; B if and only >if A = (A, B) and B > = [A, B], and that union and crosscut are associative, commutative, >and idempotent operations. > > 1940 W. V. Quine Math. Logic 56 A binary mode of statement >composition..is said to > be..idempotent if &431.<phi> &216. ;fkf&432. is true for all >statements <phi>; > > 1941 Birkhoff & MacLane Surv. Mod. Algebra xi. 313 All of these >except for the idempotent > laws and the second distributive law correspond to familiar laws of >arithmetic. > > 1941 Mind L. 274 The element is only idempotent with respect to the >combining relation > defined as the combining relation of the group. > > 1950 W. V. Quine Methods of Logic (1952) Sect.1. 3 `pp' reduces to >`p'. Conjunction is > idempotent, to persist in the jargon. > > 1959 E. M. McCormick Digital Computer Primer 181 It is further >apparent..that A + A = A > and..that A x A = A. These are sometimes referred to as the >idempotent laws. > > 1967 A. Geddes tr. Dubreil & Dubreil-Jacotin's Lect. Mod; Algebra i. >22 If every element of > E is idempotent, the composition law is called idempotent and E is >called an idempotent set. > >B. sb. An idempotent element; also in more restricted use (see quot. 1958). > > 1941 Birkhoff & MacLane Surv. Mod. Algebra i. 6 Prove that the >following rules hold in any > integral domain:..(h) the only `idempotents' (that is, elements x >satisfying xx = x) are 0 and 1. > > 1958 S. Kravetz tr. Zassenhaus's Theory of Groups (ed. 2) 182 The >element e is called an > idempotent if ee = e and if e is not a zero element. > > 1960 C. E. Rickart Gen. Theory Banach Algebras i. 35 Let &326; be a >Banach algebra and let > e be a proper idempotent in &326; (that is, e &222; 0, 1 and e2 = e). > >Hence > >idempotence > >idempotence (stress variable), > >idempotency > >idem'potency, the property of being idempotent. > > 1940 Mind XLIX. 461 The truth is that Eddington, in spite of all that >he says about getting all > the mathematics he wants out of the idempotency of the J symbols, >employs them in accordance > with the laws of ordinary algebra whenever he thinks fit. > > 1940 W. V. Quine Math. Logic 60 In the case of conjunction and >alternation, repetition of > components has..been seen to be immaterial (idempotence). > > 1957 P. Suppes Introd. Logic ix. 205 Equations (9) and (10) express >what is usually called > the idempotency of union and intersection. > > 1959 K. R. Popper Logic Sci. Discovery 351 p (aa, b) = p (a, b)... >This is the law of > idempotence, sometimes also called the `law of tautology'. > > 1960 P. Suppes Axiomatic Set Theory ii. 27 The next three theorems >assert the commutativity, > associativity, and idempotence of union. > > 1968 New Scientist 16 May 339/1 Idempotency..occurs if an operation >produces no change in > the number or set on which it operates. > I tend to agree with the line of thought that we may need to define some other term of our own to make clear what we really mean. --- Albert Lunde Albert-Lunde@nwu.eduReceived on Wednesday, 30 August 1995 09:36:52 EDT

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