Semigroups of Left I-quotients Nassraddin Ghroda May 11, 2010 - PowerPoint PPT Presentation

Semigroups of Left I-quotients Semigroups of Left I-quotients Nassraddin Ghroda May 11, 2010

Semigroups of Left I-quotients Outline Background 1 Inverse hull of left I-quotients of left ample semigroups 2 Extension of homomorphisms 3 Left I-orders in semilattices of inverse semigroups 4 Primitive inverse semigroups of left I-quotients 5

Semigroups of Left I-quotients Background Background Ore (1940)

Semigroups of Left I-quotients Background Background Ore (1940) Fountain and Petrich (1986)

Semigroups of Left I-quotients Background Background Ore (1940) Fountain and Petrich (1986) Fountain and Gould (1986)

Semigroups of Left I-quotients Background Background Ore (1940) Fountain and Petrich (1986) Fountain and Gould (1986) MacAlister (1973)

Semigroups of Left I-quotients Background Background Ore (1940) Fountain and Petrich (1986) Fountain and Gould (1986) Clifford (1953) MacAlister (1973)

Semigroups of Left I-quotients Background Background Ore (1940) Fountain and Petrich (1986) Fountain and Gould (1986) Clifford (1953) MacAlister (1973) Gould and Ghroda (2010)

Semigroups of Left I-quotients Background Left I-order Definition A subsemigroup S of an inverse semigroup Q is a left I-order in Q or Q is a semigroup of left I-quotients of S if every element of Q can be written as a − 1 b where a and b are elements of S and a − 1 is the inverse of a in the sense of inverse semigroup theory.

Semigroups of Left I-quotients Background Left I-order Definition A subsemigroup S of an inverse semigroup Q is a straight left I-order in Q or Q is a semigroup of left I-quotients of S if every element of Q can be written as a − 1 b where a R b in Q where a and b are elements of S and a − 1 is the inverse of a in the sense of inverse semigroup theory.

Semigroups of Left I-quotients Inverse hull of left I-quotients of left ample semigroups Left ample semigroups a R ∗ b if and only if xa = ya if and only if xb = yb for all x , y ∈ S 1

Semigroups of Left I-quotients Inverse hull of left I-quotients of left ample semigroups Left ample semigroups a R ∗ b if and only if xa = ya if and only if xb = yb for all x , y ∈ S 1 A semigroup S is a left ample if and only if ( i ) E ( S ) is a semilattice. ( ii ) every R ∗ -class contains an idempotent ( a R ∗ a + ). ( iii ) for all a ∈ S and all e ∈ E ( S ), ( ae ) + a = ae .

Semigroups of Left I-quotients Inverse hull of left I-quotients of left ample semigroups Left ample semigroups a R ∗ b if and only if xa = ya if and only if xb = yb for all x , y ∈ S 1 A semigroup S is a left ample if and only if ( i ) E ( S ) is a semilattice. ( ii ) every R ∗ -class contains an idempotent ( a R ∗ a + ). ( iii ) for all a ∈ S and all e ∈ E ( S ), ( ae ) + a = ae . φ : S − → I S defined by a φ = ρ a where ρ a : Sa + − → Sa defined by x ρ a = xa

Semigroups of Left I-quotients Inverse hull of left I-quotients of left ample semigroups Left ample semigroups Theorem If S is a left ample semigroup then, S is a left I-order in its inverse hull Σ( S ) ⇐ ⇒ S satisfies ( LC ) condition.

Semigroups of Left I-quotients Extension of homomorphisms Extension of homomorphisms Let S be a subsemigroup of Q and let φ : S → P be a morphism from S to a semigroup P . If there is a morphism φ : Q → P such that φ | S = φ , then we say that φ lifts to Q . If φ lifts to an isomorphism, then we say that Q and P are isomorphic over S .

Semigroups of Left I-quotients Extension of homomorphisms Extension of homomorphisms Let S be a subsemigroup of Q and let φ : S → P be a morphism from S to a semigroup P . If there is a morphism φ : Q → P such that φ | S = φ , then we say that φ lifts to Q . If φ lifts to an isomorphism, then we say that Q and P are isomorphic over S . On a straight left I-order semigroup S in a semigroup Q we define a relation T Q S on S as follows: ( a , b , c ) ∈ T Q ⇒ ab − 1 Q ⊆ c − 1 Q . S ⇐

Semigroups of Left I-quotients Extension of homomorphisms Theorem Let S be a straight left I-order in Q and let T be a subsemigroup of an inverse semigroup P. Suppose that φ : S → T is a morphism. Then φ lifts to a (unique) morphism φ : Q → P if and only if for all ( a , b , c ) ∈ S: (i) ( a , b ) ∈ R Q S ⇒ ( a φ, b φ ) ∈ R P T ; (ii) ( a , b , c ) ∈ T Q S ⇒ ( a φ, b φ, c φ ) ∈ T P T . If (i) and (ii) hold and S φ is a left I-order in P, then φ : Q → P is onto.

Semigroups of Left I-quotients Extension of homomorphisms Corollary Let S be a straight left I-order in Q and let φ : S → P be an embedding of S into an inverse semigroup P such that S φ is a straight left I-order in P. Then Q is isomorphic to P over S if and only if for any a , b , c ∈ S: ( i ) ( a , b ) ∈ R Q S ⇔ ( a φ, b φ ) ∈ R P S φ ; and ( ii ) ( a , b , c ) ∈ T Q S ⇔ ( a φ, b φ, c φ ) ∈ T P S φ .

Semigroups of Left I-quotients Extension of homomorphisms Corollary Let S be a straight left I-order in semigroups Q and P and ϕ be the embedding of S in P. Then Q ∼ = P if and only if for all a , b ∈ S, a R b in Q ⇐ ⇒ a ϕ R b ϕ and ( a , b , c ) ∈ T Q ⇒ ( a ϕ, b ϕ, c ϕ ) ∈ T P ⇐ S ϕ . S

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Semilattice of semigroups Definition Let Y be a semilattice. A semigroup S is called a semilattice Y of semigroups S α , α ∈ Y , if S = � α ∈ Y S α where S α S β ⊆ S αβ and S α ∩ S β = ∅ if α � = β .

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Strong semilattices of semigroups Definition Let Y be a semilattice. Suppose to each α ∈ Y there is associated semigroup S α and assume that S α ∩ S β � = ∅ if α � = β . For each pair α, β ∈ Y with α ≥ β, let ϕ α,β : S α − → S β be a homomorphism such that the following conditions hold: 1) ϕ α,α = ι S α , 2) ϕ α,β ϕ β,γ = ϕ α,γ if α ≥ β ≥ γ, On the set S = � α ∈ Y S α define a multiplication by a ∗ b = ( a ϕ α,αβ )( b ϕ β,αβ ) if a ∈ S α , b ∈ S β .

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Left I-orders in semilattices of inverse semigroups S = � α ∈ Y S α ( S α has (LC) and S has (LC))

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Left I-orders in semilattices of inverse semigroups S = � α ∈ Y S α ( S α has (LC) and S has (LC)) Q = � α ∈ Y Σ α ( Σ α inverse hulls of S α )

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Left I-orders in semilattices of inverse semigroups S = � α ∈ Y S α ( S α has (LC) and S has (LC)) Q = � α ∈ Y Σ α ( Σ α inverse hulls of S α ) a − 1 α b α c − 1 β d β = ( ta α ) − 1 ( rd β ) where S αβ b α ∩ S αβ c β = S αβ w and tb = rc = w for some t , r ∈ S αβ .

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Left I-orders in semilattices of inverse semigroups S = � α ∈ Y S α ( S α has (LC) and S has (LC)) Q = � α ∈ Y Σ α ( Σ α inverse hulls of S α ) a − 1 α b α c − 1 β d β = ( ta α ) − 1 ( rd β ) where S αβ b α ∩ S αβ c β = S αβ w and tb = rc = w for some t , r ∈ S αβ . Q is a semigroup

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Left I-orders in semilattices of inverse semigroups S = � α ∈ Y S α ( S α has (LC) and S has (LC)) Q = � α ∈ Y Σ α ( Σ α inverse hulls of S α ) a − 1 α b α c − 1 β d β = ( ta α ) − 1 ( rd β ) where S αβ b α ∩ S αβ c β = S αβ w and tb = rc = w for some t , r ∈ S αβ . Q is a semigroup The multiplication on Q extends the multiplication on S .

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Left I-orders in semilattices of inverse semigroups S = � α ∈ Y S α ( S α has (LC) and S has (LC)) Q = � α ∈ Y Σ α ( Σ α inverse hulls of S α ) a − 1 α b α c − 1 β d β = ( ta α ) − 1 ( rd β ) where S αβ b α ∩ S αβ c β = S αβ w and tb = rc = w for some t , r ∈ S αβ . Q is a semigroup The multiplication on Q extends the multiplication on S . S is a left I-order in Q = � α ∈ Y Σ α .

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Left I-orders in semilattices of inverse semigroups Theorem (Gantos) Let S be a semilattice of right cancellative monoids S α . Suppose that S, and S α , has (LC) condition. Then Q = � α ∈ Y Σ α is a semilattice of bisimple inverse monoids ( Σ α inverse hulls of S α ) and the multiplication in Q is defined by a − 1 α b α c − 1 β d β = ( ta α ) − 1 ( rd β ) where S αβ b α ∩ S αβ c β = S αβ w and tb = rc = w for some t , r ∈ S αβ . Corollary The semigroup S defined as above is a left I-order in Q = � α ∈ Y Σ α .

Semigroups of Left I-quotients Left I-orders in semilattices of inverse semigroups Theorem Let S = � α ∈ Y S α be a semilattice of right cancellative monoids with (LC) condition and S has (LC) condition. Let Q = � α ∈ Y Σ α where Σ α is inverse hull of S α . Then Q is a strong semilattice of monoids Σ α and S is a left I-order in Q.