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# Fundamental Concepts Of Higher Algebra

## By Nick Nicholas

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A translation, by Nick Nicholas, of the first page of _Fundamental Concepts Of
Higher Algebra_ by A. Adrian Albert (University of Chicago Press 1956).

di'e se fanva fi le pamoi bele'i papri be la'e <<lu loi jicmu sidbo pe le

CHAPTER I: GROUPS

1. _Sets and mappings._ Abstract algebra is concerned with the study of certain
mathematical objects called _algebraic systems_. Each system consists of a set
of elements, one or more operations on these elements, and a number of
assumptions (about the properties of the elements with respect to the
operations) called the _defining postulates_. In this first section we shall
introduce some of the elementary notions about sets which form the basis of
the precise definitions which we shall present of several algebraic systems.

ni'oni'oni'o 1mai cmaci girzu
ni'oni'o 1pi'e1mai selcmi ce fancu
ni'o le sucta cmacrnalgebra cu srana lenu tadni loi cmaci sibdai peme'e <<lu
cmacrnalgebra ciste li'u>>
.i ro ciste cu se pagbu lo'isu'ono selci kujo'u su'o se sumti be ri be'o
kujo'u loi selru'a {befi lei selkai {belei selci be'o} peva'o lei se sumti
be'o} neme'e <<lu satcyskicu selru'a li'u>>
.i vecu'u ledei ckupaupau cu cninyja'o so'o friseljmi sidbo {peloi selcmi
ge'uku} poi jicmu lei satci ve satcyskicu {be so'o cmacrnalgebra ciste be'o}
poi se srana da'e

Let \it A be a set whose elements a, b, c, ... are any objects whatever, and
let \it B be a second set. Then we say that B is contained in A, and write
B {subset} A, if every element of B is in A. If B {subset} A, we call B a
subset of A. We may also write A {contains} B and say that A contains B. If A
{contains} B and at least one element of A is not in B, we say that B is a
proper subset of A and write A {propercontains} B (A properly contains B), or
B {propersubset} A (B is properly contained in A). The set having no elements
is called the empty set.

ni'oca'e ge tauce'afy. ga'e .abu (to fy. sinxa la fraktur. toi) selcmi da nemu'u
nau.abu jo'u by. jo'u cy. zi'epoi cmima lu'i roda
gi ce'afy. ga'e by. selcmi gi'enaidu .abu
.iseni'ibo go
by. pagbu .abu du'i me'o by. na'u klesi .abu
gi ro cmima be by. cu cmima .abu
.i go li by. na'u klesi .abu jetnu gi by. klesi .abu
.idu'ibo {sedu'i me'o .abu na'u selkle by. lo'o} .abu selpau by.
.i go
ge li .abu na'u selkle by. jetnu gi su'o cmima be .abu naku cmima by.
gi   by. nalrolmre klesi .abu
du'i me'o .abu na'u se nalrolmemkle by.
lo'o.eme'o by. na'u nalrolmemkle .abu
.i le selcmi be noda cu se cmene <<lu le kunti selcmi li'u>>

(NOTE: relations aren't part of MEX like operators are: {du}, {klesi} (subset),
{cmima} (element of) etc. are bridi, not MEX operators. To incorporate such
relations into MEX (for example, when quoting an equation with {me'o}), {na'u}
must be used to convert the bridi into an operator. In that case, {me'o re
na'u du re} is the equation "2 = 2", and {li re na'u du re} has the evaluated
value TRUE.)

The intersection of two sets A and B is the set of all elements which are in
both A and B. We designate this set by A {intersect} B. If C {subset} A and C
{subset} B, then C is called a common subset of A and B. Every common subset
of A and B is a subset of A {intersect} B.
The union of A and B is the logical sum of A and B. It consists of all elements
which are either in A or in B. We designate it by A {union} B.

ni'o le selcmipi'i be .abu poi selcmi bei by. poi selcmi cu selcmi roda poi
cmima .abu .e by
.i le go'i cu se sinxa me'o .abu na'u selcmipi'i by
.i go li cy. na'u klesi .abu lo'o.eli cy. na'u klesi by. jetnu
gi cy. kampu klesi .abu joi by
.i ro kampu klesi be .abu joi by. cu klesi li .abu na'u selcmipi'i by.
ni'o le selcmisumji be .abu bei by. cu logji sumji .abuboi by.
.i lego'i cu se cmima roda poi cmima .abu .a by. gi'e se sinxa me'o .abu na'u
selcmisumji by.

The concepts of intersection and union may be generalised readily to several
sets. Thus if A\1,...,A\n are sets, we define their intersection A\1 {intersect}
A\2 {intersect} A\2 {intersect}...{intersect} A\n to be the set of all elements
which are simultaneously in every one of the sets A\1,...,A\n. The union A\1
{union} A\2 {union}...{union} A\n consists of all the elements in all the sets
A\1,...,A\n. Note the the equation (A\1 {intersect} A\2) {intersect} A\3 =
A\1 {intersect} (A\2 {intersect} A\3) = A\1 {intersect} A\2 {intersect} A\3
states that the set consisting of those elements in A\1 {intersect} A\2 which
are also in A\3 is the same set as that consisting of those elements of A\1
which are in A\2 {intersect} A\3 and that this set is precisely the set of
those elements which are in A\1, in A\2, and in A\3. Similarly, (A\1 {union}
A\2) {union} A\3 = A\1 {union} (A\2 {union} A\3) = A\1 {union} A\2 {union} A\3.

ni'o lesi'o selcmipi'i jo'u selcmisumji cu frili ke selsucta srana za'ure
selcmi
.i ro lu'a .abuxi1 .eli'o .abuxiny. lu'u poi selcmi zo'u
le sosyselcmipi'i be ro ri be'o no'u li selcmipi'i abuxi1 .abuxi2
.abuxi3li'o .abuxiny. cuca'e selcmi roda poi cmima role selcmi no'u .abuxi1
jo'uli'o .abuxiny.
.i le sosyselcmisumji no'u li na'u selcmisumji .abuxi1 .abuxi2 .abuxi3li'o
.abuxiny. cu selcmi roda poi cmima su'ole selcmi no'u .abuxi1 jo'uli'o .abuxiny
.i ko jundi lenu go'e
.i me'o (vei (vei .abuxi1 na'u selcmipi'i .abuxi2 ve'o) na'u selcmipi'i
.abuxi3 ve'o)
na'udu (vei .abuxi1 na'u selcmipi'i (vei .abuxi2 na'u selcmipi'i
.abuxi3 ve'o) ve'o)
na'udu na'u selcmipi'i .abuxi1 .abuxi2 .abuxi3
cu xusra lenu le selcmi be ro cmima beli .abuxi1 na'u selcmipi'i
.abuxi2 be'o poi cmima .abuxi3
cu du le selcmi be ro cmima be .abuxi1 be'o poi cmima li .abuxi2
na'u selcmipi'i .abuxi3 be'o
le selcmi be roda poi cmima .abuxi1 .e .abuxi2 .e .abuxi3
.isi'a jetnu fali (vei (vei .abuxi1 na'u selcmisumji .abuxi2 ve'o) na'u
selcmisumji .abuxi3 ve'o)
na'udu (vei .abuxi1 na'u selcmisumji (vei .abuxi2 na'u selcmisumji
.abuxi3 ve'o) ve'o)
na'udu na'u selcmisumji .abuxi1 .abuxi2 .abuxi3

se finti la .djeims. Please report any errors to me.