Nearrings whose set of N-subgroups is linearly ordered
... (c) The set E of right identities of (N, +, ∗) satisfies (C3). (d) (N, +, ∗) = W (N, ψ, Φ, E). Proof. (a) By ACC there is an integer r ≥ 1 such that Ker ψ ⊂ Ker ψ 2 ⊂ . . . ⊂ Ker ψ r−1 ⊂ Ker ψ r = Ker ψ r+1 . If Ker ψ ⊆ Im ψ r , then Ker ψ = ψ r (Ker ψ r+1 ) = ψ r (Ker ψ r ) = {0}, and (C1) (b) is f ...
... (c) The set E of right identities of (N, +, ∗) satisfies (C3). (d) (N, +, ∗) = W (N, ψ, Φ, E). Proof. (a) By ACC there is an integer r ≥ 1 such that Ker ψ ⊂ Ker ψ 2 ⊂ . . . ⊂ Ker ψ r−1 ⊂ Ker ψ r = Ker ψ r+1 . If Ker ψ ⊆ Im ψ r , then Ker ψ = ψ r (Ker ψ r+1 ) = ψ r (Ker ψ r ) = {0}, and (C1) (b) is f ...
Homological algebra
... (5) A morphism f : A → B is an isomorphism if and only if it is both mono and epi. Proposition 2.9. The last condition follows from the first four conditions. Proof. First of all, isomorphisms are always both mono and epi. The definition of an isomorphism is that it has an inverse g : B → A so that ...
... (5) A morphism f : A → B is an isomorphism if and only if it is both mono and epi. Proposition 2.9. The last condition follows from the first four conditions. Proof. First of all, isomorphisms are always both mono and epi. The definition of an isomorphism is that it has an inverse g : B → A so that ...
Dynamical systems and van der Waerden`s theorem
... Another example of vocabulary Theorem Let (X , T ) be a t.d.s. Then there is a recurrent point x. Proof sketch (for experts only): Consider the family F of closed, nonempty subsets Y of X satisfying T (Y ) ⊆ Y . Partially order the sets in F by inclusion; by Zorn’s Lemma F has a minimal element, sa ...
... Another example of vocabulary Theorem Let (X , T ) be a t.d.s. Then there is a recurrent point x. Proof sketch (for experts only): Consider the family F of closed, nonempty subsets Y of X satisfying T (Y ) ⊆ Y . Partially order the sets in F by inclusion; by Zorn’s Lemma F has a minimal element, sa ...
A survey of totality for enriched and ordinary categories
... have a Yoneda embedding Y : A -> [A°p, Set] where Set is the category of small sets - and if this embedding Y admits a left adjoint Z. Totality for these ordinary categories has been further investigated by Tholen [22], Wood [24J, and Street [18J; it turns out to imply very strong completeness ...
... have a Yoneda embedding Y : A -> [A°p, Set] where Set is the category of small sets - and if this embedding Y admits a left adjoint Z. Totality for these ordinary categories has been further investigated by Tholen [22], Wood [24J, and Street [18J; it turns out to imply very strong completeness ...
Finitely generated groups with automatic presentations
... The general idea of using finite automata to read structures is not entirely new; for example, in group theory, a group is said to be automatic if, when we code elements of the group as strings of generators, there is a regular subset L of the set of all strings of generators such that there are fi ...
... The general idea of using finite automata to read structures is not entirely new; for example, in group theory, a group is said to be automatic if, when we code elements of the group as strings of generators, there is a regular subset L of the set of all strings of generators such that there are fi ...
The Choquet-Deny theorem and distal properties of totally
... measure μ then Gμ must necessarily be an amenable subgroup [7, 30]. It follows that groups for which the theorem is valid are necessarily amenable. However, the theorem is not true for every amenable group [23]. The stronger condition, that G have polynomial growth, is sufficient for the theorem to ho ...
... measure μ then Gμ must necessarily be an amenable subgroup [7, 30]. It follows that groups for which the theorem is valid are necessarily amenable. However, the theorem is not true for every amenable group [23]. The stronger condition, that G have polynomial growth, is sufficient for the theorem to ho ...
Tannaka Duality for Geometric Stacks
... Remark 3.2. The terminology we have just introduced is borrowed from [7], with one modification: we include a hypothesis of quasi-compactness in our definition of a geometric stack. Remark 3.3. Let X be a geometric stack. Since X is quasi-compact, there exists a smooth surjection Spec A → X. Since t ...
... Remark 3.2. The terminology we have just introduced is borrowed from [7], with one modification: we include a hypothesis of quasi-compactness in our definition of a geometric stack. Remark 3.3. Let X be a geometric stack. Since X is quasi-compact, there exists a smooth surjection Spec A → X. Since t ...
Locally Finite Constraint Satisfaction Problems
... techniques used there are also quite different from ours. The line of research started in [11] was continued in [12], and certain infinite instances were studied also in [13]. In [12], [13] it is argued that infinite periodic instances naturally arise when studying large – perhaps of unknown size or ...
... techniques used there are also quite different from ours. The line of research started in [11] was continued in [12], and certain infinite instances were studied also in [13]. In [12], [13] it is argued that infinite periodic instances naturally arise when studying large – perhaps of unknown size or ...
Set theory and von Neumann algebras
... The reader may easily verify that this is equivalent to (the more geometric definition) p2 = p and ker(p)⊥ = ran(p). (Warning: From now on, when we write “projection” we will always mean an orthogonal projection. This is also the convention in most of the literature.) For projections p, q ∈ B(H), wr ...
... The reader may easily verify that this is equivalent to (the more geometric definition) p2 = p and ker(p)⊥ = ran(p). (Warning: From now on, when we write “projection” we will always mean an orthogonal projection. This is also the convention in most of the literature.) For projections p, q ∈ B(H), wr ...