The meet-semilattice congruence lattice of a frame John Frith* and Anneliese Schauerte University of Cape Town 27 September 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 1 / 20
Basics Throughout this talk L will denote a frame, the top element is denoted by 1 the bottom element is denoted by 0 . Definition A meet-semilattice congruence θ on L is an equivalence relation on L which also satisfies ( x , y ) , ( z , w ) ∈ θ ⇒ ( x ∧ z , y ∧ w ) ∈ θ . We present some well-known facts for the sake of completeness: The collection of all meet-semilattice congruences on L , Con Msl ( L ) , forms a partially ordered set under inclusion. The intersection of meet-semilattice congruences remains a meet-semilattice congruence, so meet is given by intersection. Con Msl ( L ) is a complete lattice. The top element, which we denote by ∇ , is L × L ; the bottom element, which we denote by △ , is { ( x , x ) : x ∈ L } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 2 / 20
An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 3 / 20
An example 1 • a • • b L : • 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 3 / 20
An example 1 • a • • b L : • 0 • • • • Con Msl ( L ) : • • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 3 / 20
Finite joins There is an explicit characterization of finite joins, which we need, given as follows: Suppose that θ, φ are meet-semilattice congruences on L . We say that elements x and y of L are θ − φ -linked if there is a sequence of elements x = s 0 , s 1 , s 2 . . . , s n = y of L such that, for any i ∈ { 0 , 1 , 2 , . . . , n − 1 } either ( s i , s i + 1 ) ∈ θ or ( s i , s i + 1 ) ∈ φ . We define θ ∗ φ = { ( x , y ) : x and y are θ − φ -linked } For θ, φ ∈ Con Msl ( L ) , θ ∨ φ = θ ∗ φ . Well known, we think. This extends to any finite join. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 4 / 20
Con Msl ( L ) The join of an updirected family of meet-semilattice congruences is just its union. An arbitrary join, ∨ θ i , is calculated by taking the union of all finite I joins, since these form an updirected collection. As a result, the lattice Con Msl ( L ) is compact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 5 / 20
For the sake of completeness, we recall the definition of a frame congruence: Definition A frame congruence θ on L is an equivalence relation on L which also satisfies ( x , y ) , ( z , w ) ∈ θ implies ( x ∧ z , y ∧ w ) ∈ θ . ( x i , y i ) ∈ θ for all i ∈ I implies ( ∨ x i , ∨ y i ) ∈ θ . I I The collection of all frame congruences on a frame L will be denoted by Con Frm ( L ) . It is a frame. (But the description of join given above does not apply.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 6 / 20
A structure theorem for Con Msl ( L ) . Definition For a , b ∈ L we denote by ▼ a the meet-semilattice congruence generated by the singleton { ( 0 , a ) } ∆ b the meet-semilattice congruence generated by the singleton { ( b , 1 ) } θ ab the meet-semilattice congruence generated by the singleton { ( a , b ) } . ∇ a = { ( x , y ) ∈ L × L : x ∨ a = y ∨ a } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 7 / 20
It is possible to describe ▼ a and ∆ b explicitly as follows: Lemma For a , b ∈ L 1 ▼ a = △ ∪ { ( s , t ) ∈ L × L : s , t ≤ a } ∆ b = { ( x , y ) ∈ L × L : x ∧ b = y ∧ b } . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 8 / 20
Some properties of ▼ a , etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 9 / 20
Some properties of ▼ a , etc. Lemma Let L be a frame, a , b ∈ L , { a i } i ∈ I ⊆ L . In Con Msl ( L ) we have: (a) ∧ ∧ a i . a i = ▼ I ▼ 1 (b) ▼ a ∨ ▼ b need not coincide with ▼ a ∨ b . 0 = △ and ▼ 1 = ∇ . (c) ▼ (a) ∇ a ∧ ∇ b = ∇ a ∧ b . 2 (b) ∇ a ∨ ∇ b = ∇ a ∨ b ; this does not generalize to arbitrary joins. (c) ∇ 0 = △ and ∇ 1 = ∇ . (a) ∧ ∆ a i = ∆ ∨ a i . 3 (b) ∆ a ∨ ∆ b = ∆ a ∧ b ; this does not generalize to arbitrary joins. (c) ∆ 0 = ∇ and ∆ 1 = △ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 9 / 20
Towards some structure Lemma θ ab = ( ▼ a ∧ ∆ b ) ∗ ( ▼ b ∧ ∆ a ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 10 / 20
Towards some structure Lemma θ ab = ( ▼ a ∧ ∆ b ) ∗ ( ▼ b ∧ ∆ a ) . Theorem (Structure Theorem) For any meet-semilattice congruence θ we have ∨ θ = { ( ▼ c ∧ ∆ d ) ∗ ( ▼ d ∧ ∆ c ) : ( c , d ) ∈ θ } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 10 / 20
Examples and counterexamples 1 • a • • b L : • ∇ 0 • ∇ a • •∇ b ▼ a ∨ ▼ b • Con Msl ( L ) : • • ▼ ▼ a b • △ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 11 / 20
∇ • ∆ α • ∇ β • ▼ β ∨ ▼ • ∇ γ • γ Con Msl ( L ) : ( ▼ β ∨ ▼ γ ) ∧ ∆ α ∆ γ • ▼ • • • ▼ • ∆ β β γ 1 • ∆ γ ∧ ▼ • ∇ α = ▼ • • ∆ β ∧ ▼ β γ α • γ β • α • L : • • △ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 12 / 20
Theorem In the case that L is a linear frame we claim that Con Msl ( L ) is indeed a frame but that, in general, Con Msl ( L ) ̸ = Con Frm ( L ) . (See Example below.) The proof that we found relies on the “Structure Theorem” and follows a similar route to a proof that the congruence lattice of a frame is again a frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 13 / 20
Theorem In the case that L is a linear frame we claim that Con Msl ( L ) is indeed a frame but that, in general, Con Msl ( L ) ̸ = Con Frm ( L ) . (See Example below.) The proof that we found relies on the “Structure Theorem” and follows a similar route to a proof that the congruence lattice of a frame is again a frame. Papert has this result, but it is proved differently. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frith & Schauerte (UCT) The meet-semilattice congruence lattice of a frame PWC 13 / 20
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