A first-order theory is determined by a language and a set of selected sentences of the language those sentences of the theory that are, in an arbitrary, generalized sense, the "true" ones (called the "distinguished elements" of the set).
Let (Delta _+^flat = Delta _+ times _Delta Delta ^flat ), and note that the embedding (lambda :Delta rightarrow Delta _+) extends to an embedding begin{aligned} lambda ^flat :Delta ^flat rightarrow Delta _+^flat end{aligned} (6.22 that adds a new initial element to an ordinal [n], but keeps the same distinguished element l.
As the V -monoid of a ring, M is of course conical, and contains a distinguished element (as described prior to Theorem 2).
(Bergman's Theorem) [49, Theorem 6.2] Let M be a finitely generated commutative conical monoid with distinguished element d ≠ 0, and let K be any field.
Now denote by ({text {Sets}}_+) the category of pointed sets, and consider the forgetful functor ({text {Ab}}rightarrow {text {Sets}}_+) sending an abelian group E to its underlying set with distinguished element (0 in E).
First, V ( R ) is conical: if x, y ∈ V ( R ) have x ⊕ y = 0, then x = y = 0. Second (for R unital), V ( R ) contains a distinguished element d : for each x ∈ V ( R ), there exists y ∈ V ( R ) and n ∈ N having x ⊕ y = n d (specifically, d = [ R ] ).
The association of such distinguished elements with different characteristics and the employment of suitable control techniques guarantee the voltage and frequency regulation in accordance to the international standards.
Roughly, the set theoretic definition says that a structure is an ordered n+1-tuple consisting of a set, a number of relations on this set, and a number of distinguished elements of this set.
Accompanied by vibrant drumming — Mr. Destiné collaborated for many years with the distinguished Haitian drummer Alphonse Cimber — these dances were often infused with elements of voodoo tradition.
The distinguished cast includes Christine Brandes, John Mark Ainsley, Michael Slattery and Philip Cutlip.
