From: Albert Lunde <Albert-Lunde@nwu.edu>
Date: Wed, 30 Aug 1995 11:35:26 -0600
Message-Id: <v01510100ac6a50785cbc@[129.105.110.129]>

```>>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
>      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.edu
```
Received on Wednesday, 30 August 1995 09:36:52 UTC

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