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.
(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).
This functor has a left-adjoint sending a pointed set (langle X,o rangle ) to the quotient (mathsf{Span}(X) = {mathbb {Z}}[X]/{mathbb {Z}}cdot o) of the free abelian group ({mathbb {Z}}[X]) generated by X by the subgroup spanned by the distinguished element (o in X).
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.
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).
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.
We present a general framework for the study of KMS states of generalized gauge actions on the C⁎-algebra of a Cayley graph which is pointed by considering the neutral element of the group as a distinguished vertex.
As Söderblom was a churchman and theologian as well as a distinguished historian of religion, there is without doubt an element of theological judgment influencing his stand on this matter.
