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Property of operations
Idempotence (UK: /ˌɪdɛmˈpoʊtəns/, US: /ˈaɪdəm-/) is the property of certain operations in mathematics and computer science whereby they can be applied
Idempotence
Statistical theorem
In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result that characterizes the transformation
Rao–Blackwell_theorem
Of a function, an additional effect besides returning a value
In computer science, an operation or expression is said to have a side effect if it has any observable effect other than its primary effect of reading
Side effect (computer science)
Side_effect_(computer_science)
Logical connective AND
associativity: yes distributivity: with various operations, especially with or idempotency: yes monotonicity: yes truth-preserving: yes When all inputs are true
Logical_conjunction
Idempotent linear transformation from a vector space to itself
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)
Projection_(linear_algebra)
Rule of mathematical logic
factoring in automated theorem proving systems using resolution. Known as idempotency of entailment in classical logic. Exchange, where two members on the
Structural_rule
Algebraic manipulation of "true" and "false"
failure of commutativity would then appear as a failure of symmetry. Idempotence of ∧ and ∨ can be visualized by sliding the two circles together and
Boolean_algebra
Program function without side effects
that, given a particular input, will always produce the same output Idempotence – Property of operations whereby they can be applied multiple times without
Pure_function
American mathematician and philosopher (1937–2023)
and Tierney showed that the conditions it needs to satisfy are just idempotence and the preservation of finite intersections. These Lawvere-Tierney topologies
William_Lawvere
Discrete unit of communication
implementation. However, exactly-once delivery is often achieved through idempotency mechanisms rather than true, infrastructure-level exactly-once semantics
Message
Semigroup in which every element is idempotent
the second together with idempotence. But a magma that satisfies only the identities for the first characteristic and idempotence need not be associative
Band_(algebra)
Commonly used rules of replacement in propositional logic
are: The principle of idempotency of disjunction: P ∨ P ⇔ P {\displaystyle P\lor P\Leftrightarrow P} and the principle of idempotency of conjunction: P ∧
Tautology_(rule_of_inference)
co-idempotence. An interpretation of idempotence is that: 'Idempotence means that there is no “noise” left in the smoothed data and co-idempotence means
Lulu_smoothing
Distance from zero to a number
| a | | = | a | {\displaystyle {\bigl |}\left|a\right|{\bigr |}=|a|} Idempotence (the absolute value of the absolute value is the absolute value) | −
Absolute_value
Standard representation of a mathematical object
is a mapping c:S→S such that for all s, s1, s2 ∈ S: c(s) = c(c(s)) (idempotence), s1 R s2 if and only if c(s1) = c(s2) (decisiveness), and s R c(s)
Canonical_form
Information technology system architecture
predictable latency, high throughput, strong type safety, and reliable idempotency characteristics, making it suitable for resilient large-scale distributed
Command Query Responsibility Segregation
Command_Query_Responsibility_Segregation
IT architecture separating actions and reads
command. Martin Fowler cites the pop() method of a stack as an example. Idempotence Domain-driven design Create, read, update and delete (CRUD) Meyer, Bertrand
Command–query_separation
All points in the topological closure not belonging to the interior
{\displaystyle S.} The boundary operator thus satisfies a weakened kind of idempotence. In discussing boundaries of manifolds or simplexes and their simplicial
Boundary_(topology)
= ω2 + 3 is not even. A simple application of ordinal parity is the idempotence law for cardinal addition (given the well-ordering theorem). Given an
Even_and_odd_ordinals
implementations exist. Conversely there are undesirable properties: Non-idempotence: Repeated applications can cause arbitrary drift of points. Exception
Snap_rounding
Mathematical assumptions
{\displaystyle (x\lor y)\lor z=x\lor (y\lor z)} , and the assumption of idempotence, ( x ∨ x ) = x {\displaystyle (x\lor x)=x} , the latter shown to be redundant
Minimal axioms for Boolean algebra
Minimal_axioms_for_Boolean_algebra
Open-source software platform for remote configuring and managing computers
Bash, etc.)[citation needed]. One of the guiding goals of modules is idempotency, which means that even if an operation is repeated multiple times (e
Ansible_(software)
Programming paradigm based on applying and composing functions
to that argument list (sometimes called referential transparency or idempotence), i.e., calling the pure function again with the same arguments returns
Functional_programming
System with multiple networked computers
implementation. However, exactly-once delivery is often achieved through idempotency mechanisms rather than true, infrastructure-level exactly-once semantics
Distributed_computing
Logical connective OR
∨ (b ∨ c)) ≡ ((a ∨ b) ∨ (a ∨ c)) (a ∨ (b ≡ c)) ≡ ((a ∨ b) ≡ (a ∨ c)) Idempotency: a ∨ a ≡ a Monotonicity: (a → b) → ((c ∨ a) → (c ∨ b)) (a → b) → ((a
Logical_disjunction
Element mapped to itself by a mathematical function
permutations Eigenvector Equilibrium Fixed points of a Möbius transformation Idempotence Infinite compositions of analytic functions Invariant (mathematics) Brown
Fixed_point_(mathematics)
Generators of the Clifford algebra for relativistic quantum mechanics
^{5}\right)\psi ={\begin{pmatrix}0&0\\0&I_{2}\end{pmatrix}}\psi ~.} The idempotence of the chiral projections is manifest. By slightly abusing the notation
Gamma_matrices
Function that is its own inverse
for example in context of Kramers–Wannier duality. Atbash Automorphism Idempotence ROT13 Robert Alexander Adams, Calculus: Single Variable, 2006, ISBN 0321307143
Involution_(mathematics)
Algebraic structure modeling logical operations
[dual] x ∧ (y ∧ z) = (x ∧ y) ∧ z Abbreviations UId Unique Identity Idm Idempotence Bnd Boundaries Abs Absorption law UNg Unique Negation DNg Double negation
Boolean_algebra_(structure)
Mapping equal to its square under mapping composition
projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a disk. Originally, the
Projection_(mathematics)
Type of data structure
and idempotent. The intuition behind commutativity, associativity and idempotence is that these properties are used to make the CRDT invariant under package
Conflict-free replicated data type
Conflict-free_replicated_data_type
Branch of logic
then r Idempotence of disjunction p {\displaystyle p} ⟚ ( p ∨ p ) {\displaystyle (p\lor p)} p is true is equiv. to p is true or p is true Idempotence of conjunction
Propositional_logic
posterior, posteriority • postremogeniture potis pot- compossible, idempotence, idempotent, impossible, impotence, impotency, impotent, nilpotence,
List of Latin words with English derivatives
List_of_Latin_words_with_English_derivatives
Measure of branching complexity
Schlund, Maxmilian (2011), "An extension of Parikh's theorem beyond idempotence", arXiv:1112.2864 [cs.FL] Strahler, A. N. (1952), "Hypsometric (area-altitude)
Strahler_number
Size of a set in mathematics
"representative" of that cardinality—i.e. satisfying Hume's principle, and idempotence ( | | S | | = | S | {\displaystyle \vert \vert S\vert \vert =\vert S\vert
Cardinality
Symbol connecting formulas in logic
denoted by +, if a · (b + c) = (a · b) + (a · c) for all operands a, b, c. Idempotence Whenever the operands of the operation are the same, the compound is
Logical_connective
Algebraic structure with a binary operation
Idempotence Commutative property Associative property Cancellation property OEIS sequence (labeled) OEIS sequence (isomorphism classes) Unneeded Unneeded
Magma_(algebra)
Concept in computer programming
programs are often "pure procedures". Reentrancy is not the same thing as idempotence, in which the function may be called more than once yet generate exactly
Reentrancy_(computing)
Classical averages studied in ancient Greece
\ldots x_{n})} Idempotence M ( x , x , … x ) = x {\displaystyle M(x,x,\ldots x)=x} for all x {\displaystyle x} Monotonicity and idempotence together imply
Pythagorean_means
System for reasoning about vagueness
t-norm (that is, minimum). It has the axioms of BL plus an axiom of idempotence of conjunction, and its models are called G-algebras. Product fuzzy logic
Fuzzy_logic
Matrix that, squared, equals itself
test. Any similar matrices of an idempotent matrix are also idempotent. Idempotency is conserved under a change of basis. This can be shown through multiplication
Idempotent_matrix
German mathematician (1868–1942)
spaces which fulfilled the Kuratowski closure axioms up to the axiom of idempotence. These spaces are often also called closure spaces, and Hausdorff used
Felix_Hausdorff
Operation on the subsets of a set
its own closure, that is, if x = C ( x ) . {\displaystyle x=C(x).} By idempotency, an element is closed if and only if it is the closure of some element
Closure_(mathematics)
Class of formal logics
noncontradiction, and the principle of explosion Monotonicity of entailment and idempotency of entailment Commutativity of conjunction De Morgan duality: every logical
Classical_logic
Concept in computer science
null-pointer failures. Other uses include using a Boolean field for idempotence (so subsequent calls are nops), as in the dispose pattern. public String
Guard_(computer_science)
Overview of and topical guide to logic
Directed set Equivalence relation Euclidean relation Homogeneous relation Idempotence Intransitivity Involutive relation Partial equivalence relation Partial
Outline_of_logic
Partial order with joins
Associativity x ∧ (y ∧ z) = (x ∧ y) ∧ z Commutativity x ∧ y = y ∧ x Idempotency x ∧ x = x A meet-semilattice ⟨ S , ∧ ⟩ {\displaystyle \langle S,\land
Semilattice
If and only if relation
even itself), but logical disjunction distributes over biconditional. Idempotency: No Monotonicity: No Truth-preserving: Yes When all inputs are true,
Logical_biconditional
Set whose pairs have minima and maxima
without the distributive axiom. By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets
Lattice_(order)
Excess of a non-negative real number beyond its integer part
respectively. These two definitions of fractional-part function also provide idempotence. The fractional part defined via difference from ⌊ ⌋ is usually denoted
Fractional_part
Repeated sum of a number's digits
digital root of n {\displaystyle n} in base b {\displaystyle b} . Then: Idempotence dr b ( dr b ( n ) ) = dr b ( n ) . {\displaystyle \operatorname
Digital_root
Largest open subset of some given set
Intensive: int S ⊆ S . {\displaystyle \operatorname {int} S\subseteq S.} Idempotence: int ( int S ) = int S . {\displaystyle \operatorname {int} (\operatorname
Interior_(topology)
Concurrent constraint logic programming language
leq Y, Y leq X <=> X = Y. transitivity @ X leq Y, Y leq Z ==> X leq Z. idempotence @ X leq Y \ X leq Y <=> true. The rules can be read in two ways. In the
Constraint_Handling_Rules
True when either but not both inputs are true
a field GF(2), and as in any field they obey the distributive law.) Idempotency: no Monotonicity: no Truth-preserving: no When all inputs are true, the
Exclusive_or
Axioms for defining a topology
{\mathcal {I}}}C_{i}\in {\mathfrak {S}}[\mathbf {c} ]} . Notice that, by idempotency [K3], one may succinctly write S [ c ] = im ( c ) {\displaystyle {\mathfrak
Kuratowski_closure_axioms
a\land b=b\land a} and a ∨ b = b ∨ a {\displaystyle a\lor b=b\lor a} . Idempotence: for all a ∈ L {\displaystyle a\in L} , a ∧ a = a {\displaystyle a\land
Bounded_lattice
Logic formula
differ from the "laws" of arithmetic: Absorption (idempotency) for OR: (a ∨ a) ≡ a Absorption (idempotency) for AND: (a & a) ≡ a The sign " = " (as distinguished
Propositional_formula
Reals with an extra square root of +1 adjoined
and e ∗ = 1 2 ( 1 + j ) . {\displaystyle e^{*}={\tfrac {1}{2}}(1+j).} Idempotency means that e e = e {\displaystyle ee=e} and e ∗ e ∗ = e ∗ . {\displaystyle
Split-complex_number
material implication 3 ¬ ( p ∨ p ) {\displaystyle \lnot (p\lor p)} double negation elimination 3 ¬ p {\displaystyle \lnot p} idempotency of disjunction
Connexive_logic
Computing state associated with a point in time
implementation. However, exactly-once delivery is often achieved through idempotency mechanisms rather than true, infrastructure-level exactly-once semantics
Event_(computing)
Binary operation in relational algebra
always be substituted by the same value (this is a consequence of the idempotence of the logical AND). In particular, natural join allows the combination
Join_(relational_algebra)
Algebraic structure
The operation ∧ makes L into a semigroup that satisfies the additional idempotence law a ∧ a = a. Given a homomorphism f : S → L from an arbitrary semigroup
Semigroup
Formal systems of logic that significantly differ from standard logical systems
negation elimination, and part of De Morgan's laws; Linear logic rejects idempotency of entailment as well; Paraconsistent logic (e.g., relevance logic) rejects
Non-classical_logic
Five subtopical issues in policy debate
independent of policies, which they are not. Idempotency: Is the plan or resolution redundant to the status quo? Idempotency gives clear case argument against redundancy
Stock_issues
Type of formal logic
well as associativity, commutativity, distributivity, De Morgan, and idempotence inferences (for conjunction and disjunction). Furthermore, inconsistency-robust
Paraconsistent_logic
Repeated application of an operation to a sequence
1 0 a k = e {\displaystyle \mathop {\bigstar } _{k=1}^{0}a_{k}=e} Idempotence: if a ⋆ a = a {\displaystyle a\star a=a} , then ★ k = 1 n a = a {\displaystyle
Iterated_binary_operation
Analog of Grothendieck topology
inflationary property: s ⊆ s ¯ {\displaystyle s\subseteq {\bar {s}}} idempotence: s ¯ ≡ s ¯ ¯ {\displaystyle {\bar {s}}\equiv {\bar {\bar {s}}}} preservation
Lawvere–Tierney_topology
One of five systems of modal logic
wRu)\implies vRu} , thereby conflating necessity with possibility under idempotence. In terms of Kripke semantics, S5 is characterized by frames where the
S5_(modal_logic)
Algorithmic process of solving equations
over + {\displaystyle +} ∀ u: u ∗ u {\displaystyle u*u} = u I Idempotence of ∗ {\displaystyle *} ∀ u: n ∗ u {\displaystyle n*u} = u Nl
Unification (computer science)
Unification_(computer_science)
Semiring with minimum and addition replacing addition and multiplication
Jean-Éric (1998). "Tropical semirings" (PDF). In Gunawardena, J. (ed.). Idempotency. Publications of the Newton Institute. Vol. 11. Cambridge University
Tropical_semiring
Theorem in statistics and econometrics
{X}}^{\prime }{\tilde {y}}} Where the intermediary inequalities follow from the idempotency and symmetry of the annihilator matrix. In 1907, statistician Udny Yule
Frisch–Waugh–Lovell_theorem
Skeletonized version of algebraic geometry
Jean-Eric (1998). "Tropical semirings" (PDF). In Gunawardena, J. (ed.). Idempotency. Publications of the Newton Institute. Vol. 11. Cambridge University
Tropical_geometry
Exterior algebraic map taking tensors from p forms to n-p forms
decomposition of unity, and the projector operators on the summands fulfills idempotence formulas: ( h δ ) 2 = h δ , ( δ h ) 2 = δ h {\displaystyle (h\delta )^{2}=h\delta
Hodge_star_operator
Algebraic ring that need not have additive negative elements
Gunawardena, Jeremy (1998). "An introduction to idempotency". In Gunawardena, Jeremy (ed.). Idempotency. Based on a workshop, Bristol, UK, October 3–7
Semiring
British computer scientist
often inaccurately just called idempotence, as convergence in his meaning implied both desired end-state and idempotence of an error correction operator
Mark Burgess (computer scientist)
Mark_Burgess_(computer_scientist)
Israeli-American computer scientist
2003 Hummer, W; Rosenberg, F; Oliveira, F; Eilam, T (2013). "Testing idempotence for infrastructure as code". Middleware 2013: ACM/IFIP/USENIX 14th International
Tamar_Eilam
Convex uniform honeycomb Regular map (graph theory) (up to identity and idempotency) In a classification advanced by Conway & adopted by Coxeter, stellation
List_of_regular_polytopes
Equalities for combinations of sets
L=\varnothing .} Idempotence L ∗ L = L {\displaystyle L\ast L=L} and Nilpotence L ∗ L = ∅ {\displaystyle L\ast L=\varnothing } : L ∪ L = L (Idempotence) L ∩ L
List of set identities and relations
List_of_set_identities_and_relations
Function in logic
denoted by +, if a · (b + c) = (a · b) + (a · c) for all operands a, b, c. idempotence: Whenever the operands of the operation are the same, the connective
Truth_function
Idempotent semiring endowed with a closure operator
{\displaystyle a\in A} . The above axioms define a semiring. We further require Idempotence of + {\displaystyle +} : a + a = a {\displaystyle a+a=a} for all a ∈
Kleene_algebra
Concept in order theory
y)\wedge z} (associativity), and x ∧ x = x {\displaystyle x\wedge x=x} (idempotency). Joins are defined dually with the join of x and y , {\displaystyle
Join_and_meet
Branch of metaphysics
= i x . {\displaystyle \mathbf {i} (\mathbf {i} x)=\mathbf {i} x.} (Idempotence) C7. i ( x × y ) = i x × i y , {\displaystyle \mathbf {i} (x\times y)=\mathbf
Mereotopology
Magma obeying the Latin square property
symmetric loops that satisfy x ∗ x = 1 instead of x ∗ x = x. Without idempotency, total symmetric quasigroups correspond to the geometric notion of extended
Quasigroup
Property of many systems of logic
necessary for the conclusion. Linear logic, which lacks monotonicity and idempotency of entailment. Contraction Exchange rule Substructural logic No-cloning
Monotonicity_of_entailment
Generalization of means
{\displaystyle \ M_{f}\ } is unchanged if its arguments are permuted. Idempotency: for all x , {\displaystyle \ x\ ,} the repeated average M f (
Quasi-arithmetic_mean
{\overline {\alpha ,\alpha \vdash \beta }}} Rule of contraction (or idempotency of entailment) (aka no-deleting theorem) α , α , γ ⊢ β _ {\displaystyle
List_of_rules_of_inference
{\displaystyle 1*a=a} . Notably absent from this list is the property of idempotence a ∗ a = a {\displaystyle a*a=a} ; the closest one gets is that a ∗ a
Monoidal_t-norm_logic
Estimates values in an N-dimensional matrix
z_{ij}=z} , ∀ i , j {\displaystyle i,j} then X = P Q {\displaystyle X=PQ} . Idempotency: X = K ( Z , Y ) = Z {\displaystyle X=K(Z,Y)=Z} if Y {\displaystyle Y}
Iterative proportional fitting
Iterative_proportional_fitting
to its closure: Isotonicity: Every set is contained in its closure. Idempotence: The closure of the closure of a set is equal to the closure of that
Glossary_of_general_topology
Plotkin powertheory (after Gordon Plotkin) has the following axioms: Idempotency: x ∪ x = x Commutativity: x ∪ y = y ∪ x Associativity: (x ∪ y) ∪ z =
Power_domains
Any binary relation equal to its composition with itself
yRz both true. Some authors call such an R a dense relation. Because idempotence incorporates both transitivity and the second property above, it is a
Idempotent_relation
Semiring defined over probabilities
special conditions hold (because it may violate the distributivity or idempotence in a formal way), but it is a very useful generalized semiring in practice
Viterbi_semiring
Neuromorphic data-processing model
~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall
Receptron
Function type in category theory
subject to certain axioms (commutativity, associativity, absorption and idempotency). Thus they are F-algebras of signature P x P + P x P. It is often said
F-algebra
behavior of conjunction (for example, Gödel–Dummett logic requires its idempotence) or other connectives (for example, the logic IMTL (involutive monoidal
T-norm_fuzzy_logics
Branch of type theory
{\displaystyle \cap } ) is taken modulo associativity, commutativity and idempotence. The typing rules ( → I ) {\displaystyle (\to \!\!{\text{I}})} , ( →
Intersection_type_discipline
Graph with a median for each three vertices
{\displaystyle 0} to 1 {\displaystyle 1} and with median algebras more generally: Idempotence: m ( a , a , b ) = a {\displaystyle m(a,a,b)=a} for all a {\displaystyle
Median_graph
Algebra describing information processing
) {\displaystyle \pi _{x}(R\bowtie S)=R\bowtie \pi _{x\cap y}(S)} . idempotency If x ⊆ d ( R ) {\displaystyle x\subseteq d(R)} , then R ⋈ π x ( R ) =
Information_algebra
Formulation of matroids using closure operators
{\displaystyle c\in {\text{cl}}(A\cup \{b\})} (and hence by monotonicity and idempotence in fact c ∈ cl ( A ∪ { b } ) ∖ cl ( A ) {\displaystyle c\in {\text{cl}}(A\cup
Pregeometry_(model_theory)
Polish mathematician (1944–2005)
ISSN 0024-3795. Baksalary, Jerzy K.; Baksalary, Oskar Maria (December 2000). "Idempotency of linear combinations of two idempotent matrices". Linear Algebra and
Jerzy_Baksalary
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