idempotentadj.n.

Brit.

/ˌɪdɛmˈpəʊtnt/,

/ˌʌɪdɛmˈpəʊtnt/, U.S.

/ˌaɪdɛmˈpoʊtnt/,

/ˌidɛmˈpoʊtnt/

Origin: A borrowing from Latin, combined with an English element. Etymons: Latin īdem , potent adj.¹

Etymology: < classical Latin īdem the same (see idem n.) + potent adj.¹ Compare idemfacient adj.

Mathematics, Logic, and Computing.

A. adj.

Of a quantity or element a: having the property that a × a = a, where × represents multiplication or some other binary operation; (of an operator or operation) having the property that repeated or multiple application gives the same result; of or relating to such a property.

ΘΚΠ

the world > relative properties > number > algebra > [adjective] > relating to expressions

interscendent1796

symmetrical1816

zeta-ic1840

associative1844

discriminantal1852

symmetric1853

discriminant1870

idempotent1870

interscendental1873

bilinear1886

non-trivial1901

left1926

right1930

1870 B. Peirce in Amer. Jrnl. Math. (1881) 4 104 When an expression..raised to a square or higher power..gives itself as the result, it may be called idempotent.

1915 Trans. Amer. Math. Soc. 15 424 As a group can have but one unit operator it follows that the number of maximal groups belonging to  is equal to the number of unit or idempotent operators found in a complete set of representative residues.

1941 Mind 50 274 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) §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 × A = A. These are sometimes referred to as the idempotent laws.

1967 A. Geddes tr. P. Dubreil & M.-L. Dubreil-Jacotin 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.

2003 W. Crawford & J. Kaplan J2EE Design Patterns v. 101 An idempotent action can be repeated as many times as necessary without ill effects.

B. n.

An idempotent element.

ΘΚΠ

the world > relative properties > number > mathematical number or quantity > [noun] > particular qualities > other

digitusa1398

argumentc1405

geodeticala1690

known quantity1702

amicable number1743

summability1900

idempotent1903

modularity1927

repunit1964

palindrome1972

seed1972

1903 Trans. Amer. Math. Soc. 4 274 There are other sets of these forms..which involve the compound idempotents λ₁₁₀ + λ₂₂₀, etc.

1941 G. Birkhoff & S. 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.

1960 C. E. Rickart Gen. Theory Banach Algebras i. 35 Let  be a Banach algebra and let e be a proper idempotent in  (that is, e ≠ 0, 1 and e² = e).

2001 T. Y. Lam First Course Noncommutative Rings (ed. 2) vii. 308 To understand the structure of..rings, it is often important to study the behaviour of their idempotents.

This entry has been updated (OED Third Edition, November 2010; most recently modified version published online March 2022).

单词	idempotent
释义	idempotentadj.n. Brit. /ˌɪdɛmˈpəʊtnt/, /ˌʌɪdɛmˈpəʊtnt/, U.S. /ˌaɪdɛmˈpoʊtnt/, /ˌidɛmˈpoʊtnt/ Origin: A borrowing from Latin, combined with an English element. Etymons: Latin īdem , potent adj.¹ Etymology: < classical Latin īdem the same (see idem n.) + potent adj.¹ Compare idemfacient adj. Mathematics, Logic, and Computing. A. adj. Of a quantity or element a: having the property that a × a = a, where × represents multiplication or some other binary operation; (of an operator or operation) having the property that repeated or multiple application gives the same result; of or relating to such a property. ΘΚΠ the world > relative properties > number > algebra > [adjective] > relating to expressions interscendent1796 symmetrical1816 zeta-ic1840 associative1844 discriminantal1852 symmetric1853 discriminant1870 idempotent1870 interscendental1873 bilinear1886 non-trivial1901 left1926 right1930 1870 B. Peirce in Amer. Jrnl. Math. (1881) 4 104 When an expression..raised to a square or higher power..gives itself as the result, it may be called idempotent. 1915 Trans. Amer. Math. Soc. 15 424 As a group can have but one unit operator it follows that the number of maximal groups belonging to  is equal to the number of unit or idempotent operators found in a complete set of representative residues. 1941 Mind 50 274 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) §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 × A = A. These are sometimes referred to as the idempotent laws. 1967 A. Geddes tr. P. Dubreil & M.-L. Dubreil-Jacotin 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. 2003 W. Crawford & J. Kaplan J2EE Design Patterns v. 101 An idempotent action can be repeated as many times as necessary without ill effects. B. n. An idempotent element. ΘΚΠ the world > relative properties > number > mathematical number or quantity > [noun] > particular qualities > other digitusa1398 argumentc1405 geodeticala1690 known quantity1702 amicable number1743 summability1900 idempotent1903 modularity1927 repunit1964 palindrome1972 seed1972 1903 Trans. Amer. Math. Soc. 4 274 There are other sets of these forms..which involve the compound idempotents λ₁₁₀ + λ₂₂₀, etc. 1941 G. Birkhoff & S. 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. 1960 C. E. Rickart Gen. Theory Banach Algebras i. 35 Let  be a Banach algebra and let e be a proper idempotent in  (that is, e ≠ 0, 1 and e² = e). 2001 T. Y. Lam First Course Noncommutative Rings (ed. 2) vii. 308 To understand the structure of..rings, it is often important to study the behaviour of their idempotents. This entry has been updated (OED Third Edition, November 2010; most recently modified version published online March 2022). < adj.n.1870
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