# Download Algebra Some Current Trends by Avramov L.L. (ed.), Tchakerian K.B. (ed.) PDF

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

Similar algebra books

Algebra: A Text-Book of Determinants, Matrices, and Algebraic Forms

A few of the earliest books, quite these relationship again to the 1900s and sooner than, are actually super scarce and more and more dear. we're republishing those vintage works in cheap, top of the range, smooth variants, utilizing the unique textual content and art.

Extra resources for Algebra Some Current Trends

Sample text

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 .