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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
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)
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
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)
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
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
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
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
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
Concept in computer programming
are often "pure procedures". Reentrancy should not be confused with idempotence, in which the function may be called more than once yet generate exactly
Reentrancy_(computing)
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
Discrete unit of communication
implementation. However, exactly-once delivery is often achieved through idempotency mechanisms rather than true, infrastructure-level exactly-once semantics
Message
Algebraic structure with a binary operation
Idempotence Commutative property Associative property Cancellation property OEIS sequence (labeled) OEIS sequence (isomorphism classes) Unneeded Unneeded
Magma_(algebra)
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
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)
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)
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)
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
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
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
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
implementations exist. Conversely there are undesirable properties: Non-idempotence: Repeated applications can cause arbitrary drift of points. Exception
Snap_rounding
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
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)
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
= 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
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
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
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
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
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
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
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)
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
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
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)
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)
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)
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
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
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)
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)
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
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
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
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
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)
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
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
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
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)
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)
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
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
Type of formal logic
well as associativity, commutativity, distributivity, De Morgan, and idempotence inferences (for conjunction and disjunction). Furthermore, inconsistency-robust
Paraconsistent_logic
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
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
Theorem in statistics and econometrics
{X}}^{\prime }{\tilde {y}}} Where the intermediary steps follow from the idempotency and symmetry of the annihilator matrix. In 1907, statistician Udny Yule
Frisch–Waugh–Lovell_theorem
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
material implication 3 ¬ ( p ∨ p ) {\displaystyle \lnot (p\lor p)} double negation elimination 3 ¬ p {\displaystyle \lnot p} idempotency of disjunction
Connexive_logic
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
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)
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
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
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)
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
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
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
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
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
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)
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
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
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
{\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
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
{\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
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
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
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
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
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
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
Plotkin powertheory (after Gordon Plotkin) has the following axioms: Idempotency: x ∪ x = x Commutativity: x ∪ y = y ∪ x Associativity: (x ∪ y) ∪ z =
Power_domains
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
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
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
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
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
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)
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
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
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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