| 释义 |
idempotent, a. and n.|aɪˈdɛmpəʊtənt, ˌaɪdəmˈpəʊtənt|
[f. L. idem same + potent-em powerful, potent]
A. adj. Of a quantity or element a: having the property that a × a = a, where × 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.
1870B. 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. 1937A. A. Albert Mod. Higher Algebra (1938) iii. 88 A matrix E is called idempotent if E2 = E. 1937Duke Math. Jrnl. III. 629 We recall that A ⊃ B if and only if A = (A, B) and B = [A, B], and that union and crosscut are associative, commutative, and idempotent operations. 1940W. V. Quine Math. Logic 56 A binary mode of statement composition..is said to be..idempotent if ⌜ϕ ≡ .ϕκϕ⌝ is true for all statements ϕ. 1941Birkhoff & 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. 1941Mind L. 274 The element is only idempotent with respect to the combining relation defined as the combining relation of the group. 1950W. V. Quine Methods of Logic (1952) §1. 3 ‘pp’ reduces to ‘p’. Conjunction is idempotent, to persist in the jargon. 1959E. M. McCormick Digital Computer Primer 181 It is further apparent..that A + A = A and..that A × A = A. These are sometimes referred to as the idempotent laws. 1967A. 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. n. An idempotent element; also in more restricted use (see quot. 1958).
1941Birkhoff & 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. 1958S. 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. 1960C. E. Rickart Gen. Theory Banach Algebras i. 35 Let 𝔘 be a Banach algebra and let e be a proper idempotent in 𝔘 (that is, e ≠ 0, 1 and e2 = e).
Hence idempotence (stress variable), idemˈpotency, the property of being idempotent.
1940Mind 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. 1940W. V. Quine Math. Logic 60 In the case of conjunction and alternation, repetition of components has..been seen to be immaterial (idempotence). 1957P. Suppes Introd. Logic ix. 205 Equations (9) and (10) express what is usually called the idempotency of union and intersection. 1959K. 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’. 1960P. Suppes Axiomatic Set Theory ii. 27 The next three theorems assert the commutativity, associativity, and idempotence of union. 1968New Scientist 16 May 339/1 Idempotency..occurs if an operation produces no change in the number or set on which it operates. |