The original is at http://www.lojban.org/files/texts/algebra The below is me translating the lojban only back to english, without reference to the original English. My commentary is in [[...]]. Note that I've read a *lot* of texts like this before, so I may not be the best choice to be doing this; I'd like to see someone with little mathematical training give it a shot. - --------- di'e se fanva fi le pamoi bele'i papri be la'e <> pefi'e la'o <> itu'e This next is a translation of the first of the pages of "The Basic Ideas Of Advanced Algebra", written by A. Adrian Albert. [[I have no idea what the itu'e is for; maybe for the di'e to bind to?]] 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 <> .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 <> .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 Chapter 1: Mathematical Groups Section 1.1: Set Functions [[I have no idea what the "ce" is for there.]] Abstract algebra is relevant to the study of mathematical objects called "Algebraic Systems". All systems are composed of at least zero units [[selci??]] along with at least one function of those units along with some postulates about the properties of the units within the context of those functions, which we call "Defining Postulates". In this chapter we introduce several known [[?? friseljmi]] ideas about sets which form the basis of exact definitions of several alegbraic systems which we will need later. 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 <> I define both uppercase Fracture font A [[?? what's with tau, and what's fraktur? Perhaps he meant http://mathworld.wolfram.com/Doublestruck.html]] as the set with members such as [[ !!ERROR!! The nau causes the sentence to not parse, and makes little sense anyways. My best guess is that na'a was intended, but na'o, na'u and nai are all possibilities. Of those, only na'a and nai make the sentence parse. I assume na'a. ]] a, b, and c, which are members of the set of all such things, [[I find poi cmima lu'i roda very confusing; perhaps he meant the set of all things period, but in that case da should not have been re-used]] and uppercase Fracture font B is a set and is not equal to A. Therefore IFF B is a part of A, also shown as as B {subset} A, then all members of B are members of A. IFF B {subset} A is true then B is a subset of A. [[Note that "{subset}" is supposed to be the subset symbol; see http://mathworld.wolfram.com/Subset.html, so that sentence is expository rather than redundant.]] Equally, it is shown by A {superset} B that A is a whole which contains B. IFF both A {superset} B is true and at least some members of A are not members of B, then B is a non-total subset of A, or A {proper superset} B and B {proper subset} A. The set of nothing is named "the empty set". 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. The set multiplication of set A and set B is the set of all elements of A and B. It is shown as A {intersect} B. IFF C {subset} A and C {subset} B is true then C is a common subset of A and [[joi seems wrong here, as it implies, to me, the union of the two sets.]] B. All common subsets of A and B are subsets of A {intersect} B. 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 set addition of A and B is the logical addition of A and B. It is the set of all members of A or B and is show as A {union} B. 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 The idea of intersection and union is easily abstracted across more than 2 sets. Given that all of A_1 and ... A_n are sets, the many-intersection of all of them, or [[I bet there was supposed to be a na'u here, since it doesn't parse without it. Also, to parse there needs to be a boi between abu and xi *every* *single* *time*; that should be an *easy* fix in the parser.]] {big-intersect} A_1 A_2 A_3 ... A_n is defined as the set of all members of all the sets A_1 .. A_n. Many-union, or {big-union} A_1 A_2 A_3 ... A_n is the set of all members of at least one of the sets A_1 ... A_n. Please pay attention to how that worked. ((A_1 {intersect} A_2) {intersect} A_3) = (A_1 {intersect} (A_2 {intersect} A_3)) = [[I'm *sure* sosyselcmipi'i was meant here, so I will take it as such.]] {big-intersect} A_1 A_2 A_3 asserts that [[strange use of xusra; where's the agent? seems like nibli would work better. And shouldn't x2 be a du'u?]] the set of all members of A_1 {intersect} A_2 which are members of A_3 is equal [[should be dunli, not du, IMO, but that's a nitpick; I don't like du as a selbri]] to the set of all members of A_1 which are members of A_2 {intersect} A_3 [[and here it turns out that du was used because it has an arbitrary number of places, so ignore me]] which is equal to the set of all members of A_1 and A_2 and A_3. Similarily, it is true that ((A_1 {union} A_2) {union} A_3) = (A_1 {union} (A_2 {union} A_3)) = [[Again, I'm sure he meant sosyselcmisumji here]] {big-union} A_1 A_2 A_3.