Categorical-data.pptx for the college students on a specific grade level

Instruction: Choose the word that is appropriately describe in the given meaning inside the
envelope. The words are in the box below.
Group 1:
Is the branch of mathematics that studies algebraic systems and the manipulation
of equations within those systems. It is a generalization of arithmetic that
includes variables besides regular numbers and algebraic operations other than the
standard arithmetic operations like addition and multiplication.
Group 2:
Studies algebraic structures, which consist of a set of mathematical objects together with
one or several binary operations defined on that set. It is a generalization of elementary
and linear algebra since it allows mathematical objects other than numbers and non-
arithmetic operations.
Pedagogy Abstract Algebra Homomorphism Algebraic Structure Categories

This article is about a branch of
mathematics. For the Swedish
band, see Abstrakt Algebra.
"Modern algebra" redirects here.
For van der Waerden's book,
Moderne Algebra.

Abstrakt Algebra
• was a Swedish experimental metal band with influences
from power metal and doom metal.
• It was founded by bassist Leif Edling in 1994, shortly after his main
project Candlemass split up.
• They made one album, but Edling had already started working on a
second album with a different line-up.
• However, due to the commercial failure of Abstrakt Algebra, Edling
reformed Candlemass while taking with him some of the ideas for
that second album, as well as drummer Jejo Perkovic. There,
materialised on the Dactylis Glomerata album.
• And as such Abstrakt Algebra was over. That second album,
called Abstrakt Algebra II, was later included as a bonus disc in the
2006 re-release of Dactylis Glomerata.

Moderne Algebra is a two-volume German textbook
on graduate abstract algebra by Bartel Leendert van
der Waerden (1930, 1931), originally based on
lectures given by Emil Artin in 1926 and by Emmy
Noether (1929) from 1924 to 1928. The English
translation of 1949–1950 had the title Modern
algebra, though a later, extensively revised edition in
1970 had the title Algebra.

In 1975 van der Waerden described the sources he drew upon to write
the book.
In 1997 Saunders Mac Lane recollected the book's influence:
•Upon its publication it was soon clear that this was the way that
algebra should be presented.
•Its simple but austere style set the pattern for mathematical texts in
other subjects, from Banach algebras to topological group theory.
•[Van der Waerden's] two volumes on modern algebra ... dramatically
changed the way algebra is now taught by providing a decisive example
of a clear and perspicacious presentation. It is, in my view, the most
influential text of algebra of the twentieth century.

In mathematics, more
specifically algebra, abstract
algebra or modern algebra is the study
of algebraic structures.[1] Algebraic
structures
include groups, rings, fields, modules, vecto
r spaces, lattices, and algebras over a field.

In algebraic structure consists of a
nonempty set A (called the underlying
set, carrier set or domain), a collection
of operations on A (typically binary
operations such as addition and multiplication),
and a finite set of identities, known as axioms,
that these operations must satisfy.

An algebraic structure may be based on other
algebraic structures with operations and
axioms involving several structures. For
instance, a vector space involves a second
structure called a field, and an operation
called scalar multiplication between elements
of the field (called scalars), and elements of
the vector space (called vectors).

Abstract algebra is the name that is commonly
given to the study of algebraic structures. The
general theory of algebraic structures has been
formalized in universal algebra. Category
theory is another formalization that includes also
other mathematical
structures and functions between structures of the
same type (homomorphisms).

In universal algebra, an algebraic structure is called an algebra;[1] this
term may be ambiguous, since, in other contexts, an algebra is an
algebraic structure that is a vector space over a field or a module over
a commutative ring.
The collection of all structures of a given type (same operations and
same laws) is called a variety in universal algebra; this term is also
used with a completely different meaning in algebraic geometry, as an
abbreviation of algebraic variety. In category theory, the collection of
all structures of a given type and homomorphisms between them form
a concrete category.

Algebraic structures, with their
associated homomorphisms, form mathematical
categories. Category theory gives a unified framework
to study properties and constructions that are similar
for various structures.
Universal algebra is a related subject that studies types
of algebraic structures as single objects. For example,
the structure of groups is a single object in universal
algebra, which is called the variety of groups.

In algebra, a homomorphism is a structure-
preserving map between two algebraic structures of the same
type (such as two groups, two rings, or two vector spaces).
The word homomorphism comes from the Ancient Greek
language: ὁμός (homos) meaning "same"
and μορφή (morphe) meaning "form" or "shape". However,
the word was apparently introduced to mathematics due to a
(mis)translation of German ähnlich meaning "similar"
to ὁμός meaning "same".[1] The term "homomorphism"
appeared as early as 1892, when it was attributed to the
German mathematician Felix Klein (1849–1925).

Homomorphisms of vector spaces are also called linear
maps, and their study is the subject of linear algebra.
The concept of homomorphism has been generalized,
under the name of morphism, to many other structures that
either do not have an underlying set, or are not algebraic.
This generalization is the starting point of category theory.
A homomorphism may also be an isomorphism,
an endomorphism, an automorphism, etc. (see below).
Each of those can be defined in a way that may be
generalized to any class of morphisms.

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