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By Avramov L.L. (ed.), Tchakerian K.B. (ed.)

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X ∗ y = y ∗ x. 2. (x ∗ y) ∗ z = x ∗ (y ∗ z). 3. x ∗ x = x. An operation ∗ on S that is commutative, associative, and idempotent is called a semilattice operation on S. 2) (L, ∧, ∨) is an algebra with two binary operations ∧ and ∨ that are both commutative, associative, and idempotent, hence are semilattice operations. If one omits the operation ∨ from a lattice, one is left with a semilattice (L, ∧); and if one omits the operation ∧, one is left with a semilattice (L, ∨). 2 A bisemilattice is an algebra (B, ∧, ∨) with two binary operations, both of which are semilattice operations.

But any increasing function above f ∨ g lies above f L and g L , hence above f L ∨ g L . So f L ∨ g L is the least increasing function above f ∨ g. The second equation is similar. We remark that the equations (f ∧ g)L = f L ∧ g L and (f ∧ g)R = f R ∧ g R need not hold (Exercise 4). However, since (f ∧ g)L ≤ f L and (f ∧ g)L ≤ g L , we see that (f ∧ g)L ≤ f L ∧ g L and similarly (f ∧ g)R ≤ f R ∧ g R . Our final two lemmas provide relationships among the convolution operations ⊓ and ⊔, and L and R.

Proof. 5 states that f ⊓ g = (f ∧ g R ) ∨ (f R ∧ g) = (f ∨ g) ∧ f R ∧ g R So if f = f ⊓ g, then f R ∧ g ≤ f ≤ g R . Conversely, if f R ∧ g ≤ f ≤ g R , then f ⊓ g = (f ∧ g R ) ∨ (f R ∧ g) = f ∨ (f R ∧ g) = f so f ⊑⊓ g. Item 2 follows similarly. To conclude this section, we collect various properties of these orders. 4 The following hold for f, g ∈ M: 1. f ⊑⊓ 11 and 10 ⊑⊔ f . 2. f ⊑⊓ g if and only if g ∗ ⊑⊔ f ∗ . 3. If f and g are decreasing, then f ⊑⊓ g if and only if f ≤ g. 4. If f is decreasing, then f ⊑⊓ g if and only if f ≤ g R .

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