[lemma=□] □
[lemma=algebra] □
[lemma=be] □
As a type hierarchy becomes more complex it becomes more and more difficult to model it precisely with a class hierarchy as shown in the computer algebra example . All algebras A considered here are assumed to be finite dimensional as a vector space over their coefficient field K , and to possess a multiplicative identity element 1A . This is not the place for details ( see [ Goguen , 2003 ] for that ) , but we can say that hidden algebra provides a precise way to handle both the display and implementation aspects of examples like that described above , and the corresponding extension of semiotic morphisms gives a precise basis for comparing the quality of interface designs realizing the desired dynamics , without bias towards any particular implementation . T he PCMI sponsors four intensive math classes ranging from basic math to Algebra during a six-week summer program for 6th to 8th grade students . Solving for the unknown atmospheric quantities then becomes a linear algebra problem where the solution vector is equal to the inverse of the kernel matrix multiplied by the set of independent measurements . Yes , Linear Algebra provides a beautiful geometric picture . Algebra provides tools and ways of thinking that are necessary for solving problems in a wide variety of disciplines , such as science , business , social sciences , fine arts , and technology . We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A ^ e-modules .