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Topics referred to by the same term
The term normal closure is used in two senses in mathematics: In group theory, the normal closure of a subset of a group is the smallest normal subgroup
Normal_closure
Smallest normal group containing a set
In group theory, the normal closure of a subset S {\displaystyle S} of a group G {\displaystyle G} is the smallest normal subgroup of G {\displaystyle
Normal_closure_(group_theory)
Type of algebraic field extension
contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:
Normal_extension
Operation on the subsets of a set
language. In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set. In mathematical
Closure_(mathematics)
Topics referred to by the same term
(disambiguation) Normal closure (disambiguation) Normal Field (disambiguation) Normal form (disambiguation) Normal order (disambiguation) Normal Township (disambiguation)
Normal
Subgroup invariant under conjugation
itself, so the normal subgroups are precisely the kernels of homomorphisms with domain G . {\displaystyle G.} Normalizer Normal closure Normal core Malnormal
Normal_subgroup
Computer network protocol
Start closing handshake. Optional code: If specified, must be 1000 (Normal closure) or in the range 3000 to 4999 (application-defined). Defaults to 1000
WebSocket
Mathematical group
the orientations of blocks. This group is a normal subgroup of G. It can be represented as the normal closure of some moves that flip a few edges or twist
Rubik's_Cube_group
Cryptographic protocols for securing data in transit
is missing before the session closure, the session may be resumed automatically (with its handshakes). Normal closure of a session after termination
Transport_Layer_Security
Algebraic field extension
splitting field of x 2 − 2 {\displaystyle x^{2}-2} ; the second has normal closure that includes the complex cubic roots of unity, and so is not a splitting
Galois_extension
Technique for creating lexically scoped first class functions
In programming languages, a closure, also lexical closure or function closure, is a technique for implementing lexically scoped name binding in a language
Closure (computer programming)
Closure_(computer_programming)
Metabolic disorder leading to high bone density and malformation
fingers (the distal phalanges) to be unusually short; and delays the normal closure of the connections (sutures) of the skull bones in infancy, so that
Pycnodysostosis
Group of eye diseases related to poor retinal and nerve perfusion
(increased cup-to-disc ratio on fundoscopic examination). Acute angle-closure glaucoma, a medical emergency due to the risk of impending permanent vision
Glaucoma
Specification of a mathematical group by generators and relations
{\displaystyle F_{S}} by the smallest normal subgroup that contains each element of R. (This subgroup is called the normal closure N of R in F S {\displaystyle
Presentation_of_a_group
Saving scheme by the Government of India
interest at the prevailing rate. If the girl is over 18 and married, normal closure is allowed. The Sukanya Samriddhi Yojana (SSY) scheme is highly flexible
Sukanya_Samriddhi_Account
Surgical modification of the eyelids
eyelid anatomy is essential for preserving eyelid support, maintaining normal closure and achieving natural, balanced results. The eyelid consists of multiple
Blepharoplasty
normal closure The normal closure of a subset S of a group G is the intersection of all normal subgroups of G that contain S. normal core The normal core
Glossary_of_group_theory
Construction of a larger algebraic field by "adding elements" to a smaller field
algebraic extension F/K admits a normal closure L, which is an extension field of F such that L / K {\displaystyle L/K} is normal and which is minimal with this
Field_extension
Generalization of the Riemann zeta function for algebraic number fields
factorization using the normal closure of the larger field and induced representations. Let: M {\displaystyle M} be the normal closure of L / K {\displaystyle
Dedekind_zeta_function
Quotient space of a codomain of a linear map by the map's image
is the quotient of H by the normal closure of the image of f. In the case of abelian groups, since every subgroup is normal, the cokernel is just H modulo
Cokernel
distribution with a normal distribution. In 2006, Singh and Hespanha proposed a closure which approximates the population distribution as a log-normal distribution
Moment_closure
Group presentations useful in knot theory
group is trivial. The group is finitely presented. The group is the normal closure of a single generator. Conditions (3) and (4) are essentially the Wirtinger
Wirtinger_presentation
S2CID 119348804 Murty, V. Kumar (2001), "Class numbers of CM-fields with solvable normal closure", Compositio Mathematica, 127 (3): 273–287, doi:10.1023/A:1017589432526
List of number fields with class number one
List_of_number_fields_with_class_number_one
{\displaystyle H^{K}} . Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups. Here are some facts relating
Paranormal_subgroup
Group in group theory mathematics
than the property of being FC: every subgroup has finite index in its normal closure. Scott (1987), 15.1.1, p. 441. Scott (1987), 15.1.2, p. 441. Scott,
FC-group
Vertical transport
be closed until a few seconds after opening. In this case (hastening normal closure), the door close button has no effect. However, the door close button
Elevator
Any of certain special normal subgroups of a group
the concept of normal core is that of normal closure which is the smallest normal subgroup of G containing H. Normal cores are important in the context of
Core_(group_theory)
Every polynomial has a real or complex root
Since the normal closure of K over R still has a finite degree over C (or R), we may assume without loss of generality that K is a normal extension of
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Aspect of category theory
group homomorphisms, their coequalizer is the quotient of Y by the normal closure of the set S = { f ( x ) g ( x ) − 1 ∣ x ∈ X } {\displaystyle S=\{f(x)g(x)^{-1}\mid
Coequalizer
subgroup is a subgroup whose normal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal only if it is the whole group
Contranormal_subgroup
Well-quasi-ordering of finite trees
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Kruskal's_tree_theorem
transitive closure of the relation of normality. If every subnormal subgroup of G is normal in G, then G is called a T-group. Characteristic subgroup Normal core
Subnormal_subgroup
Mathematical element
ring are integral over Z. In geometry, integral closure is closely related with normalization and normal schemes. It is the first step in resolution of
Integral_element
Special types of subgroups encountered in group theory
apply to normalizers. The normalizer should not be confused with the normal closure. Clearly C G ( S ) ⊆ N G ( S ) {\displaystyle C_{G}(S)\subseteq N_{G}(S)}
Centralizer_and_normalizer
Category whose objects are groups and whose morphisms are group homomorphisms
cokernel (given by the quotient group of H {\displaystyle H} by the normal closure of f ( G ) {\displaystyle f(G)} in H {\displaystyle H} ). Unlike in
Category_of_groups
Order-preserving mathematical function
extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure Topology & Orders Alexandrov topology
Monotonic_function
Type of group in mathematics
{\displaystyle N_{2}=\langle \langle s\rangle \rangle _{F(X)}} be the normal closures of r and s in F(X) accordingly. Then N 1 = N 2 {\displaystyle N_{1}=N_{2}}
One-relator_group
Disruption or damage to living tissue
thoroughly evaluated by a healthcare professional for proper management. In normal physiology, all wounds will undergo a series of steps collectively known
Wound
Town in Illinois, United States
Normal is a town in McLean County, Illinois, United States. The population was 52,736 at the 2020 census. Normal is the smaller of two principal cities
Normal,_Illinois
Theoretical model of computation
recursive function. More specifically it aims to define rigorously head normal form reduction of a lambda term using call-by-name reduction. Thanks to
Krivine_machine
Smallest transitive relation containing a given binary relation
In mathematics, the transitive closure R+ of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive
Transitive_closure
In algebra, the integral closure of an ideal I {\displaystyle I} of a commutative ring R {\displaystyle R} , denoted by I ¯ {\displaystyle {\overline
Integral_closure_of_an_ideal
element w ∈ F(A) we have w = 1 in G if and only if w belongs to the normal closure of R in F(A) that is, if and only if w can be represented as w = u 1
Van_Kampen_diagram
Type of Dirichlet series associated to number field extensions
relies on inductivity property. Denote: M {\displaystyle M} be for normal closure for L / K {\displaystyle L/K} H {\displaystyle H} be Galois group for
Artin_L-function
Upper airway closure disrupting sleep
alcohol) that may relax their body tone excessively and interfere with normal arousal from sleep mechanisms. The special symptom of OSA syndrome in adults
Obstructive_sleep_apnea
Group theory function
in G. Note that this is equivalent to saying that w belongs to the normal closure of R in F(X), that is, there exists a representation of w as w = u 1
Dehn_function
crisis caused by the war between Iran and the U.S.-Israel coalition. The closure of the Strait of Hormuz, through which around 20% of the world's oil trade
2026_Iran_war_fuel_crisis
Abnormal vocal sound due to excess airflow through the nose during speech
resulting from an incomplete closure of the soft palate and/or velopharyngeal sphincter (velopharyngeal insufficiency). In normal speech, nasality is referred
Hypernasal_speech
Subset of a preorder that contains all larger elements
preordered set ( X , ≤ ) , {\displaystyle (X,\leq ),} the upper closure or upward closure of x {\displaystyle x} is defined by ↑ x = { u ∈ X : x ≤ u } {\displaystyle
Upper_and_lower_sets
The 2026 Iran war, including the closure of the Strait of Hormuz, has led to what the International Energy Agency has characterized as the "largest supply
Economic impact of the 2026 Iran war
Economic_impact_of_the_2026_Iran_war
In mathematics, the reflexive closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest reflexive relation on X {\displaystyle
Reflexive_closure
Axioms in topology defining notions of "separation"
that the closure of H is contained in G. X is fully normal if every open cover has an open star refinement. X is fully T4, or fully normal Hausdorff
Separation_axiom
word-hyperbolic. If R and S are finite symmetrized subsets of F(X) with equal normal closures in F(X) such that both presentations ⟨ X ∣ R ⟩ {\displaystyle \langle
Small_cancellation_theory
Algebraic structure
algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if
Integrally_closed_domain
Formal system for transcribing expressions into equivalent terms
is sufficient to describe important properties of rewriting systems like normal forms, termination, and various notions of confluence. Historically, there
Abstract_rewriting_system
Platelet function analyser
membrane. A normal Col/Epi closure time (<180 seconds) excludes the presence of a significant platelet function defect. If the Col/Epi closure time is prolonged
PFA-100
1968 film
Norman Normal is a 1968 animated cartoon short, produced by Warner Bros.-Seven Arts Animation. It was produced as a collaboration between musician Paul
Norman_Normal
Human heart defect present at birth
blood flows between the atria (upper chambers) of the heart. Some flow is a normal condition both pre-birth and immediately post-birth via the foramen ovale;
Atrial_septal_defect
Algebraic correspondence
{\displaystyle L\otimes _{R}S} is a normal ring. Huneke, Craig; Swanson, Irena (2006), "Ch. 19", Integral closure of ideals, rings, and modules, London
Normal_homomorphism
Valve in the human heart between the left ventricle and the aorta
aortic valve to close. The closure of the aortic valve contributes the A2 component of the second heart sound (S2). Closure of the aortic valve permits
Aortic_valve
Defunct private high school in Asheville, North Carolina
Booker T. Washington and W.E.B. DuBois, the Allen School did function as a normal school. But it also offered a variety of classes, including ones related
Allen School (Asheville, North Carolina)
Allen_School_(Asheville,_North_Carolina)
complete sublattice in the lattice of all normal subgroups of G. Characteristic subgroup Normal closure Normal core de Giovanni, F.; Russo, A.; Vincenzi
Serial_subgroup
Algebraic structure
extensions of intuitionistic logic and normal extensions of the modal logic S4.Grz. Given an interior algebra A, the closure operator obeys the axioms of the
Interior_algebra
Closure that seals the top of a bottle
with normal application torque. Pressures sensitive liners are supplied pressed into closures and adhere to the bottle finish when the closure is applied
Bottle_cap
television show debuts, finales, cancellations, and channel launches, closures and rebrandings, as well as information about controversies and carriage
2026_in_Philippine_television
Medical condition
other complications. In infancy, phimosis is considered physiological (normal). At birth, the foreskin is naturally adhered to the glans, and cannot be
Phimosis
Set of isolated points in the spectrum of an operator with finite-rank Riesz projectors
projector is finite. The discrete spectrum can also be defined as the set of normal eigenvalues. A point λ ∈ C {\displaystyle \lambda \in \mathbb {C} } in the
Discrete spectrum (mathematics)
Discrete_spectrum_(mathematics)
Commercial airport near Bloomington, Illinois, USA
Bloomington–Normal (IATA: BMI, ICAO: KBMI, FAA LID: BMI) is a public airport in McLean County, Illinois, three miles east of Bloomington and southeast of Normal.
Central Illinois Regional Airport
Central_Illinois_Regional_Airport
Music club night in Los Angeles (2006–2018)
2011. Low End Theory Website "Los Angeles' Low End Theory announces its closure". Mixmag. Retrieved July 10, 2018. "Low End Theory says goodbye with guests
Low_End_Theory
VIP membership past May 27, 2011 may be eligible for a refund due to the closure. To find out more and to see if they qualify for a refund, gamers could
JamLegend
\oplus \{T(A)\mid L\vdash A\},} where ⊕ {\displaystyle \oplus } denotes normal closure. It can be shown that every superintuitionistic logic also has a largest
Modal_companion
In mathematics, the symmetric closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest symmetric relation on X {\displaystyle
Symmetric_closure
Web browser developed by Google
wrapper fitted around the normal installer, and many businesses find that this arrangement does not meet their needs. The normal downloaded Chrome installer
Google_Chrome
the topology on X is locally convex then the closure of a normal cone is a normal cone. If C is a normal cone in X and B is a bounded subset of X then
Ordered topological vector space
Ordered_topological_vector_space
Noise generated by the heart
sound is intensified due to the increased cardiac output. Emily’s heartbeat Normal heart sounds of a 16 year old girl at rest, as heard with a stethoscope
Heart_sounds
American multinational technology company
Sergio (2021). "Working from home and income inequality: risks of a 'new normal' with COVID-19". Journal of Population Economics. 34 (1): 303–360. doi:10
British actor and comedian (born 1965)
Coogan co-founded the production company Baby Cow Productions with Henry Normal. In 2013, he co-wrote, produced, and starred in the film Philomena, which
Steve_Coogan
Notation for context-free formal grammars
formal language theory, a context-free grammar, G, is said to be in Chomsky normal form (first described by Noam Chomsky) if all of its production rules are
Chomsky_normal_form
father. She later meets with Butcher, telling him she sought to give him closure. In season five, Connie has died and Sam lives in pain with his cancer
List_of_The_Boys_characters
11th episode of the 7th season of The X-Files
"Closure" is the eleventh episode of the seventh season of the science fiction television series The X-Files, and the 150th episode overall. It was directed
Closure_(The_X-Files)
Relational database theory concept
FDs. We are going to calculate a closure for A (written as A+) from this relationship. A → B B → C AB → D The closure would be as follows: A → A (by Armstrong's
Functional_dependency
On chains and antichains in partial orders
Theorem 5.6, p. 60, ISBN 0-387-24219-8, MR 2127991. Lovász, László (1972), "Normal hypergraphs and the perfect graph conjecture", Discrete Mathematics, 2 (3):
Dilworth's_theorem
Japanese army prison camp
occasionally even alcohol, with the intent of keeping prisoners in their normal state of health at the beginning of experiments. In many cases, prisoners
Zhongma_Fortress
High school in Many, Louisiana (1957–1970)
listings in Sabine Parish, Louisiana Sabine Parish School Board Sabine Normal and Industrial Institute "Sabine High School, Many, LA". November 18, 2018
Sabine High School (Louisiana)
Sabine_High_School_(Louisiana)
Concept in algebraic geometry
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally
Normal_scheme
passed, will allow her to seek the ZBZ presidency again. Kappa Tau faces closure due to housing violations until they agree to host the honor engineering
List_of_Greek_episodes
2020, and their underestimation of that support was within the realm of a normal polling error. Going into the election, most polls showed the race to be
2024 United States presidential election
2024_United_States_presidential_election
Online battle royale video game
providing random items which may otherwise not be able to be found during normal gameplay. At the completion of each round, players will receive in-game
Rules_of_Survival
2025 Indian film
she loved and expresses a desire to offer them some form of emotional closure. Acting on this, Prabhendu later travels to Kuttikkanam and meets Maya’s
Sarvam_Maya
integral closure is a finite module.[citation needed] This prompted Zariski (1948) to ask whether a local Noetherian domain such that its integral closure is
Analytically_unramified_ring
Variant in human skulls
with persistent metopic suture). "Benign metopic ridge from a normal physiological of closure can be present in 10–25% of infants (Cohen and MacLean, 2000)"
Metopic_ridge
Capacity Distributor) (in Hungarian). Retrieved 6 August 2024. "Track closure news" (PDF) (in Hungarian). Retrieved 8 September 2025. "Track lock information"
List of railway lines in Hungary
List_of_railway_lines_in_Hungary
McDonald's restaurant chain
planet, CosMcland. It also spoke in a mixture of robot-esque sounds and normal speech. In 2023, McDonald's started focusing more on its classic mascots
CosMc's
Indian actress and former Chief Minister of Tamil Nadu (1948–2016)
200 units, gave 750 units of power to power loom weavers, implemented closure of 500 liquor shops and reduction of working hours of liquor outlets emergence
J._Jayalalithaa
School in Essex County, Virginia, US (1902–1948)
Industrial, Normal, and Collegiate Institute (1903), Bowling Green Industrial Academy (1903), King William Academy (1903), Fredericksburg Normal and Industrial
Rappahannock Industrial Academy
Rappahannock_Industrial_Academy
Small domesticated carnivorous mammal
study. Some concerns of early neutering were metabolic, retarded physeal closure, and urinary tract disease related. About 250 heritable genetic disorders
Cat
Special subset of a partially ordered set
of terminology. For such posets, downward direction and upward closure reduce to: Closure under finite intersections If A, B ∈ F, then so too is A ∩ B ∈
Filter_(mathematics)
Medical condition
Intraoperative transesophageal echocardiography is used to confirm secure closure of the VSD, normal function of the aortic and tricuspid valves, good ventricular
Ventricular_septal_defect
American musician, songwriter and actor (1958–2016)
Library—the country's first full-service library for African-Americans—from closure. That same year, he anonymously paid off the medical bills of drummer Clyde
Prince_(musician)
Private therapeutic boarding school in Poland, Maine, United States
as deeply traumatic to her, and a direct cause of her inability to form normal relationships with her family members and other people. Students would occasionally
Élan_School
NORMAL CLOSURE
NORMAL CLOSURE
Biblical
treasurer of Nergal
Girl/Female
American, Australian, British, Chinese, Christian, Danish, English, Finnish, French, German, Latin, Swedish
From the North; Pattern; Courage; Norseman; Rule; Standard; Female Version of Norman
Boy/Male
Assamese, Bengali, Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Punjabi, Sikh, Sindhi, Tamil, Telugu, Traditional
Kindness; Clean; Pure; Talent Person; The One who is Pure
Boy/Male
American, Australian, French, Scottish
From the Northern Town
Girl/Female
Indian
Soft
Boy/Male
French Teutonic American English German
From the north.
Surname or Lastname
English, Irish (Ulster), Scottish, and Dutch
English, Irish (Ulster), Scottish, and Dutch : name applied either to a Scandinavian or to someone from Normandy in northern France. The Scandinavian adventurers of the Dark Ages called themselves norðmenn ‘men from the North’. Before 1066, Scandinavian settlers in England were already fairly readily absorbed, and Northman and Normann came to be used as bynames and later as personal names, even among the Saxon inhabitants. The term gained a new use from 1066 onwards, when England was settled by invaders from Normandy, who were likewise of Scandinavian origin but by now largely integrated with the native population and speaking a Romance language, retaining only their original Germanic name.French : regional name for someone from Normandy.Dutch : ethnic name for a Norwegian.Jewish (Ashkenazic) : variant of Nordman.Jewish : Americanized form of some like-sounding Ashkenazic name.Swedish : from norr ‘north’ + man ‘man’.Albert Andriessen Bradt, a settler in Rensselaerswijck on the upper Hudson River in NY, was originally from Norway and was known as de Norrman (‘the Norwegian’). The waterway south of Albany which powered his mills became known as the Normanskill (‘the Norman’s Waterway’), by which name it is still known today.
Boy/Male
Biblical
Treasurer of Nergal.
Girl/Female
Indian, Punjabi, Sikh, Telugu
Pure; Without Any Impurity
Female
English
English name derived from the gem name, from Latin corallium, probably ultimately from Hebrew goral, CORAL means "small pebble."
Male
Scottish
Scottish form of Irish Gaelic Cormac, CORMAG means "son of defilement."
Male
English
English form of Norwegian Normund, NORMAND means "north protection."
Male
English
English form of Teutonic Nordemann, NORMAN means "northman."
Girl/Female
Latin American
Rule; pattern. Can also be a feminine form of Norman: from the North.
Boy/Male
Afghan, Arabic
Handsome
Boy/Male
Scottish American
From the north valley.
Female
Italian
 Italian name invented by Felice Romani in his libretto for Belini's opera of the same name, derived from Latin norma, NORMA means "standard, rule." Compare with another form of Norma.
Boy/Male
Hindu
Clean, Pure
Female
English
 Feminine form of English Norman, NORMA means "northman." Compare with another form of Norma.
Boy/Male
Shakespearean
Hamlet, Prince of Denmark' Fortinbras, Prince of Norway.
NORMAL CLOSURE
NORMAL CLOSURE
Boy/Male
Hindu, Indian
Emotion Full
Girl/Female
Australian, French, Hebrew, Jewish
The Light is Mine; I have Light; The Lord is My Light
Girl/Female
Tamil
Muktakesha | à®®à¯à®•à¯à®¤à®•à¯‡à®·à®¾
One who has open tresses
Girl/Female
Tamil
Suneela is a common Hindu female, Deep, Dark blue color , Extending all over as the blue Sky
Female
Native American
Native American Cheyenne name AYASHE means "little one."
Girl/Female
Arabic, Indonesian, Islamic, Malaysian, Muslim, Pakistani, Urdu
Friendship; Love; Harmony
Girl/Female
Latin
Patient.
Biblical
the bed of Lord; the Lord hath taken away; poverty,whom Jehovah nourishes
Boy/Male
Arabic, Muslim
Pride of the Chiefs
Female
English
Variant spelling of English Kayley, KALEIGH means "slender."
NORMAL CLOSURE
NORMAL CLOSURE
NORMAL CLOSURE
NORMAL CLOSURE
NORMAL CLOSURE
adv.
In a normal manner.
a.
Human; belonging to man, who is mortal; as, mortal wit or knowledge; mortal power.
a.
Pertaining to, or situated near, the back, or dorsum, of an animal or of one of its parts; notal; tergal; neural; as, the dorsal fin of a fish; the dorsal artery of the tongue; -- opposed to ventral.
a.
Denoting certain hypothetical compounds, as acids from which the real acids are obtained by dehydration; thus, normal sulphuric acid and normal nitric acid are respectively S(OH)6, and N(OH)5.
n.
See Wormil.
a.
Sound; normal.
a.
Of or pertaining to Normandy or to the Normans; as, the Norman language; the Norman conquest.
a.
Northern; pertaining to the north, or to the north wind; as, a boreal bird; a boreal blast.
a.
Denoting that series of hydrocarbons in which no carbon atom is united with more than two other carbon atoms; as, normal pentane, hexane, etc. Cf. Iso-.
n.
See Mormal.
a.
According to a square or rule; perpendicular; forming a right angle. Specifically: Of or pertaining to a normal.
a.
Both renal and portal. See Portal.
a.
Done in due form, or with solemnity; according to regular method; not incidental, sudden or irregular; express; as, he gave his formal consent.
n.
The quality, state, or fact of being normal; as, the point of normalcy.
a.
Alt. of Loral
a.
According to an established norm, rule, or principle; conformed to a type, standard, or regular form; performing the proper functions; not abnormal; regular; natural; analogical.
a.
Serving to teach or convey a moral; as, a moral lesson; moral tales.
n.
See Wormil.
a.
Not according to rule; abnormal.
a.
Having the form or appearance without the substance or essence; external; as, formal duty; formal worship; formal courtesy, etc.