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However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.
In instrumental music, a style of playing that imitates the way the human voice might express the music, with a measured tempo and flexible legato. cantilena a vocal melody or instrumental passage in a smooth, lyrical style canto Chorus; choral; chant cantus mensuratus or cantus figuratus (Lat.) Meaning respectively "measured song" or "figured ...
Musical symbols are marks and symbols in musical notation that indicate various aspects of how a piece of music is to be performed. There are symbols to communicate information about many musical elements, including pitch, duration, dynamics, or articulation of musical notes; tempo, metre, form (e.g., whether sections are repeated), and details about specific playing techniques (e.g., which ...
Definition Lacuna: gap: A silent pause in a piece of music Ossia: from o ("or") + sia ("that it be") A secondary passage of music which may be played in place of the original Ostinato: stubborn, obstinate: A repeated motif or phrase in a piece of music Pensato: thought out: A composed imaginary note Ritornello: little return
The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
Consequently, bounded completeness is equivalent to the existence of all non-empty infima. A poset is a complete lattice if and only if it is a cpo and a join-semilattice. Indeed, for any subset X, the set of all finite suprema (joins) of X is directed and the supremum of this set (which exists by directed completeness) is equal to the supremum ...
Alternatively, if the meet defines or is defined by a partial order, some subsets of indeed have infima with respect to this, and it is reasonable to consider such an infimum as the meet of the subset. For non-empty finite subsets, the two approaches yield the same result, and so either may be taken as a definition of meet.
You are correct. The limit inferior and limit superior should be defined in terms of limit points. Limit inferior and limit superior are more general terms that represent the infimum and supremum (respectively) of all limit points of a set. The limit inferior and limit superior of a sequence (or a function) are specializations of this definition.