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12 added 47 characters in body

Regular. To start off:

The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.

A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.

A point $\zeta_0$ on the boundary of a domain in $\mathbb C$ is called regular if there exists a subharmonic barrier function $b(z)$ defined within $D$ near $\zeta$. This may not be the standard definition but Gamelin's complex Analysis defines it as a subharmonic function $\omega(z)$ on ${|z-\zeta_0|<\delta}\cap D$ which is negative everywhere, tends to 0 at $\zeta_0$, but $\limsup(\omega(z))<0$ as $z$ tends to any other boundary point of $D$ within distance $\delta$ of $\zeta_0$.

I've borrowed/paraphrased the following from the Wikipedia disambiguation page but removed a couple that either are not too relevant to pure math or qualify the "regularity" more. Feel free to put them in too.

Regular cardinal, a cardinal number that is equal to its cofinality

Regular category

Regular element, and regular sequence and regular immersion.

Regular code, an algebraic code with a uniform distribution of distances between codewords

Regular graph, a graph such that all the degrees of the vertices are equal

The regularity lemma, which has nothing to do with regular graphs

Regular polygon, and regular polyhedron

Regular prime: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $\mathbb Q[\zeta_p]$.

Regular surface in algebraic geometry

Regularity of an elliptic operator

JS Milne's comment: A regular map is a morphism of algebraic varieties.

Regular value of a differentiable map

Regular ring (Note: this definition can be made noncommutative. A right noetherian ring R is said to be right regular if every finitely generated right R-module has finite global dimension. See Lam's Lectures in Modules and Rings, Section 5G.)

(von Neumann) Regular ring

Regular language, a language that can be accepted by a finite state machine.

Absolutely regular is a synonym for $\beta$-mixing in stochastic processes.

Regular matroid, a matroid which is representable over every field. In this sense, all graphs are regular (their cycle matroids are regular), which has nothing to do with regular graphs.

Regular. To start off:

The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.

A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.

A point $\zeta_0$ on the boundary of a domain in $\mathbb C$ is called regular if there exists a subharmonic barrier function $b(z)$ defined within $D$ near $\zeta$. This may not be the standard definition but Gamelin's complex Analysis defines it as a subharmonic function $\omega(z)$ on ${|z-\zeta_0|<\delta}\cap D$ which is negative everywhere, tends to 0 at $\zeta_0$, but $\limsup(\omega(z))<0$ as $z$ tends to any other boundary point of $D$ within distance $\delta$ of $\zeta_0$.

I've borrowed/paraphrased the following from the Wikipedia disambiguation page but removed a couple that either are not too relevant to pure math or qualify the "regularity" more. Feel free to put them in too.

Regular cardinal, a cardinal number that is equal to its cofinality

Regular category

Regular element, and regular sequence and regular immersion.

Regular code, an algebraic code with a uniform distribution of distances between codewords

Regular graph, a graph such that all the degrees of the vertices are equal

The regularity lemma, which has nothing to do with regular graphs

Regular polygon, and regular polyhedron

Regular prime: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $\mathbb Q[\zeta_p]$.

Regular surface in algebraic geometry

Regularity of an elliptic operator

JS Milne's comment: A regular map is a morphism of algebraic varieties.

Regular value of a differentiable map

Regular ring (Note: this definition can be made noncommutative. A right noetherian ring R is said to be right regular if every finitely generated right R-module has finite global dimension. See Lam's Lectures in Modules and Rings, Section 5G.)

(von Neumann) Regular ring

Regular language, a language that can be accepted by a finite state machine.

Absolutely regular is a synonym for $\beta$-mixing in stochastic processes.

Regular matroid, a matroid which is representable over every field. In this sense, all graphs are regular (their cycle matroids are regular).

Regular. To start off:

The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.

A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.

A point $\zeta_0$ on the boundary of a domain in $\mathbb C$ is called regular if there exists a subharmonic barrier function $b(z)$ defined within $D$ near $\zeta$. This may not be the standard definition but Gamelin's complex Analysis defines it as a subharmonic function $\omega(z)$ on ${|z-\zeta_0|<\delta}\cap D$ which is negative everywhere, tends to 0 at $\zeta_0$, but $\limsup(\omega(z))<0$ as $z$ tends to any other boundary point of $D$ within distance $\delta$ of $\zeta_0$.

I've borrowed/paraphrased the following from the Wikipedia disambiguation page but removed a couple that either are not too relevant to pure math or qualify the "regularity" more. Feel free to put them in too.

Regular cardinal, a cardinal number that is equal to its cofinality

Regular category

Regular element, and regular sequence and regular immersion.

Regular code, an algebraic code with a uniform distribution of distances between codewords

Regular graph, a graph such that all the degrees of the vertices are equal

The regularity lemma, which has nothing to do with regular graphs

Regular polygon, and regular polyhedron

Regular prime: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $\mathbb Q[\zeta_p]$.

Regular surface in algebraic geometry

Regularity of an elliptic operator

JS Milne's comment: A regular map is a morphism of algebraic varieties.

Regular value of a differentiable map

Regular ring (Note: this definition can be made noncommutative. A right noetherian ring R is said to be right regular if every finitely generated right R-module has finite global dimension. See Lam's Lectures in Modules and Rings, Section 5G.)

(von Neumann) Regular ring

Regular language, a language that can be accepted by a finite state machine.

Absolutely regular is a synonym for $\beta$-mixing in stochastic processes.

9 added 'regular language' from theoretical computer science.; deleted 3 characters in body
8 Cleaned up regular topology
7 added 143 characters in body
6 added 440 characters in body
5 Added regular value of differentiable maps