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Double Groups and Semigroups Science Atlantic 2014 University of New Brunswick, Saint John Darien DeWolf Dalhousie University October 4, 2014 Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 1 / 42 Double

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Double Groups and Semigroups

Science Atlantic 2014 University of New Brunswick, Saint John Darien DeWolf Dalhousie University October 4, 2014

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Double Groups

Definition

A double group (G, ⊚, ⊙) is a set G equipped with two group operations ⊙ and ⊚ that satisfy the the middle-four interchange law: for all a, b, c, d ∈ S, (a ⊚ b) ⊙ (c ⊚ d) = (a ⊙ c) ⊚ (b ⊙ d).

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Observation 1:

(G, ⊚, ⊙) a double group. Let 1⊙ be the identity for ⊙ and 1⊚ the identity for ⊚. 1⊙ = 1⊙ ⊙ 1⊙ = (1⊙ ⊚ 1⊚) ⊙ (1⊚ ⊚ 1⊙) = (1⊙ ⊙ 1⊚) ⊚ (1⊚ ⊙ 1⊙) = 1⊚ ⊚ 1⊚ = 1⊚ Observation 1: The identities of a double group must agree.

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Observation 2:

(G, ⊚, ⊙) a double group. Let 1 be the (shared) identity for ⊙ and ⊚. a ⊙ b = (a ⊚ 1) ⊙ (1 ⊚ b) = (a ⊙ 1) ⊚ (1 ⊙ b) = a ⊚ b Observation 2: The operations of a double group must agree.

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Observation 3:

(G, ⊚, ⊙) a double group. Let 1 be the (shared) identity for ⊙ and ⊚ and write products by concatenation. ab = (1a)(b1) = (1b)(a1) = ba Observation 3: The operations of a double group must agree and must be commutative. Eckmann-Hilton Argument: Double groups are essentially Abelian groups.

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Double Semigroups

Definition

A double semigroup (S, ⊚, ⊙) is a set equipped with two associative binary

perations satisfying the middle-four interchange law: for all a, b, c, d ∈ S,

(a ⊚ b) ⊙ (c ⊚ d) = (a ⊙ c) ⊚ (b ⊙ d). Horizontal product: a ⊚ b = . Vertical product: a ⊙ b = . Middle-four:

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Example

Any set D can be made into a double semigroup by equipping it with left and right projection: a ⊙ b = a a ⊚ b = b. Associative: Middle-four interchange law:

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Theorem

For any sixteen elements a, b, ... in any double semigroup, this equation holds: (The empty boxes represent fourteen nameless elements, that are the same

n each side of the equation, and in the same order.)

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Double Cancellative Semigroups

Definition

A semigroup S is said to be right cancellative if, for any a, b, c ∈ S, ac = bc implies a = b. left cancellative if, for any a, b, c ∈ S, ca = cb implies a = b. cancellative if both left cancellative and right cancellative. A double semigroup is said to be cancellative if both of its operations are.

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Corollary

A cancellative double semigroup D is commutative.

Proof.

Suppose that a, b ∈ D. Let c ∈ D be any element of D. Then by Theorem 4, and thus, by the definition of cancellative,

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Proposition

If (S, ⊙, ⊚) is a double cancellative semigroup, then ⊙ = ⊚.

Proof.

Let a, b ∈ S and consider the following sequence of tile slidings, where each blank square is some nameless semigroup element:

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Definition

Two elements x and y in a semigroup S are said to be inverse if x = xyx and y = yxy. A semigroup is said to be an inverse semigroup if every element has a unique inverse. A double semigroup is said to be inverse if both of its operations are.

Theorem (Kock)

Double inverse semigroups are commutative.

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Need a lemma to prove this:

Lemma

Let S be a double inverse semigroup. Then the inverse operations of S

commute. That is, a⊙⊚ = a⊚⊙ for all a ∈ S.

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a⊙ = a⊙ ⊚ a⊚⊙ ⊚ a⊙?

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a⊙ = a⊙ ⊚ a⊚⊙ ⊚ a⊙? Yes.

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Recall: a⊙ = a⊙ ⊚ a⊚⊙ ⊚ a⊙ In particular, for a⊚ : a⊚⊙ = a⊚⊙ ⊚ a⊙ ⊚ a⊚⊙ That is, a⊙⊚ = a⊚⊙

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Proof of Commutativity.

Fact:

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Proof of Commutativity.

Fact:

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Proof of Commutativity.

Fact:

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Proof of Commutativity.

Fact:

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Proof of Commutativity.

Fact:

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Proof of Commutativity.

Fact:

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Proof of Commutativity.

Similarly, one calculates that The vertical inverse of a ⊚ b is a⊙ ⊚ b⊙ ⊚ a⊚⊙ ⊚ b⊙⊚ ⊚ a⊙b⊙. Repeat to show: The vertical inverse of b ⊚ a is also a⊙ ⊚ b⊙ ⊚ a⊚⊙ ⊚ b⊙⊚ ⊚ a⊙b⊙. This implies: a ⊚ b = b ⊚ a

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It can be shown that

Theorem

Double inverse semigroups are essentially commutative inverse semigroups.

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