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Two-Variable Weighted Shifts in Multivariable Operator Theory
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
When Mz is (jointly) bounded below, the map φ : ℤn → Y given by φ(α) (a) ≔ a(α) (α ∈ ℤn, a ∈ Y) is injective and open, and X ≔ φ(ℤ+n)− ⊆ Y is compact. Thus, X is a suitable compactification of ℤ+n [41, Lemmas 2.1 and 2.3]. If G: = Y × ℤn|X ≔ {(y, α) ∈ Y × ℤn : y ∈ X and y + α ∈ X}, we see that G is the groupoid obtained by reducing the transformation group Y × ℤn to X (X then becomes the unit space of G). Analysis of X leads to a description of the ideal structure of C*(Mz), based on the correspondence between open invariant subsets of X and closed ideals in C*(Mz).
Natural Numbers
Published in Nita H. Shah, Vishnuprasad D. Thakkar, Journey from Natural Numbers to Complex Numbers, 2020
Nita H. Shah, Vishnuprasad D. Thakkar
Definition 1.50: Set A with binary operation ⊕ is called a groupoid or magma.
Elements of Algebra
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
To be completely precise in denoting a semigroup, we should use some symbolism such as (X,°,=), which specifies the set of elements, the binary relation, and the equality relation used to specify the equality of elements; i.e., x°(y°z)=(x°y)°z. However, it is customary to use either the pair (X,°) or simply the letter designation of the set of elements, in this case X, as a designation of the groupoid or semigroup, provided there is no danger of confusion as to the notation being used for binary composition. Furthermore, algebraists, as a rule, do not use a special symbol such as ∘ to denote a binary operation different from the usual addition or multiplication. They stick with the conventional additive or multiplicative notation and even call these operations addition and multiplication. We follow this convention to some extent by using the notation x+y when viewing x°y as an additive operation and x·y or x×y, or even xy, when viewing x°y as a multiplicative operation. There is a sort of gentlemen's agreement that the zero symbol 0 is used to denote an additive identity and the symbol 1 to denote a multiplicative identity, even though they may not actually be denoting the integers 0 and 1. Of course, if a person is also talking about numbers at the same time, other symbols are used to denote these identities in order to avoid confusion.
The class of states of the world as an
Published in International Journal of General Systems, 2018
Fernando Tohmé, Gianluca Caterina, Rocco Gangle
An alternative characterization shows that an -groupoid G is an -category. Besides objects, G has morphisms of all orders (a morphism of order is a morphism between morphisms of order k) and morphisms of order k are iso for .6