Double Groups and Semigroups Science Atlantic 2014 University of - PowerPoint PPT Presentation

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 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 ) . Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 2 / 42

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. Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 3 / 42

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. Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 4 / 42

Observation 3: ( G , ⊚ , ⊙ ) a double group. Let 1 be the (shared) identity for ⊙ and ⊚ and write products by concatenation. ab = (1 a )( b 1) = (1 b )( a 1) = ba Observation 3: The operations of a double group must agree and must be commutative. Eckmann-Hilton Argument: Double groups are essentially Abelian groups. Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 5 / 42

Double Semigroups Definition A double semigroup ( S , ⊚ , ⊙ ) is a set equipped with two associative binary operations 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: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 6 / 42

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: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 7 / 42

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 on each side of the equation, and in the same order.) Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 8 / 42

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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. Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 22 / 42

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, Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 23 / 42

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: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 24 / 42

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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. Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 30 / 42

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 . Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 31 / 42

a ⊙ = a ⊙ ⊚ a ⊚ ⊙ ⊚ a ⊙ ? Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 32 / 42

a ⊙ = a ⊙ ⊚ a ⊚ ⊙ ⊚ a ⊙ ? Yes. Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 33 / 42

Recall: a ⊙ = a ⊙ ⊚ a ⊚ ⊙ ⊚ a ⊙ In particular, for a ⊚ : a ⊚ ⊙ = a ⊚ ⊙ ⊚ a ⊙ ⊚ a ⊚ ⊙ That is, a ⊙ ⊚ = a ⊚ ⊙ Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 34 / 42

Proof of Commutativity. Fact: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 35 / 42

Proof of Commutativity. Fact: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 36 / 42

Proof of Commutativity. Fact: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 37 / 42

Proof of Commutativity. Fact: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 38 / 42

Proof of Commutativity. Fact: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 39 / 42

Proof of Commutativity. Fact: Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 40 / 42

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 Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 41 / 42

It can be shown that Theorem Double inverse semigroups are essentially commutative inverse semigroups. Darien DeWolf Dalhousie University Double Groups and Semigroups October 4, 2014 42 / 42