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In On Utumi's ring of quotients, Canad. J. Math. 15(1963), 363-370, J. Lambek says:

As a matter of historical record, the minimal injective extension of a module is a special case of the "algebraic closure" of an algebraic system considered by K. Shoda in his paper Zur Théorie der algebraischen Erweiterungen, Osaka Math. J., 4 (1952), 133-144.

Since I cannot read German, I cannot understand what the paper by Shoda says, although I think that he does defer the definitions to his previous papers Uber die allgemeinen algebraischen Systeme I-VIII (links provided below).

I would like to know:

1) What exactly his algebraic systems are (maybe the varieties of universal algebra?).

2) Which are his fundamental results about said systems (in these or other papers).

A published translation/review/summary would serve me perfectly. The MathSciNet reviews of the first papers are too vague, enumerating examples and talking about "certain operations", "various relations", etc.

Links to the I-VIII papers:

1) https://projecteuclid.org/download/pdf_1/euclid.pja/1195578672

2) https://projecteuclid.org/download/pdf_1/euclid.pja/1195573978

3) https://projecteuclid.org/download/pdf_1/euclid.pja/1195573940

4) https://projecteuclid.org/download/pdf_1/euclid.pja/1195573901

5) https://projecteuclid.org/download/pdf_1/euclid.pja/1195573626

6) https://projecteuclid.org/download/pdf_1/euclid.pja/1195573484

7) https://www.jstage.jst.go.jp/article/pjab1912/19/9/19_9_515/_pdf

8) https://www.jstage.jst.go.jp/article/pjab1912/20/8/20_8_584/_pdf

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  • $\begingroup$ It would help to have the earlier definitions. My knowledge of German is poor compared to many other members of this forum. However, on first glance the vibe I get is that it is universal algebraic in character, with the notion of algebraically independent elements added as extensions. If you provide the links to the background papers, I can firm up this guess and translate a few of the rough bits for you. Gerhard "Understands Universal Algebra In English" Paseman, 2017.10.09. $\endgroup$
    – Gerhard Paseman
    Commented Oct 9, 2017 at 19:44
  • $\begingroup$ @GerhardPaseman Sure! I just added the links (all of them have free access). Please, if you finally translate something, do it as a formal answer! $\endgroup$
    – Jose Brox
    Commented Oct 9, 2017 at 20:46
  • $\begingroup$ I will look over a few of them, and see about answering your questions. In the spirit of efficiency, please indicate some passages which you think may hold the most light, and I will prioritize those and give my translation of them (with no guarantee) to the best of my ability . Gerhard "Best Given Time And Energy" Paseman, 2017.10.09. $\endgroup$
    – Gerhard Paseman
    Commented Oct 9, 2017 at 21:23
  • $\begingroup$ @GerhardPaseman That sounds great! It is really difficult to choose without understanding anything, but: 1) Paper I: two first paragraphs of section 1 (definition of system), two first paragraphs of section 2 (definition of primitive system and the following proposition), first paragraph of section 3 (definition of free system). 2) Paper II: Section 8. 3) Paper "Zur theorie..." section 1 (definition of algebraic and formulation of propositions), theorem about the algebraic closure cited by Lambek (don't know which one, should be like injective extension), theorems on section 2 (polynomials) $\endgroup$
    – Jose Brox
    Commented Oct 9, 2017 at 22:50
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    $\begingroup$ I have read some of it. I do not have a simple answer, because there are some subtleties I am not grasping. I found I.5 helpful, as he is using these special terms as binary operations. The meromorphism between equivalence class algebras seems new to me, as it is not quite isomorphism of quotients. A lot of it is familiar though, and I hope to post on I ,II, and IV, and then see if I myself can answer your questions. Gerhard "Thank You For The Reminder" Paseman, 2017.10.24. $\endgroup$
    – Gerhard Paseman
    Commented Oct 24, 2017 at 19:06

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I have decided to not give a short answer to this question.

Although much of Shoda's work seems familiar, there are some twists that make it slightly different from the algebraic systems (universal algebras or algebraic structures) I have studied. Much as I would like to point to recent texts on Universal Algebra and say "you can get the concepts there", Shoda's emphasis and choices suggest something which I do not see clearly. So I will attempt to summarize enough of the foundations to try to express the main results. In particular, I will highlight what I think are the main points to grasp, in hopes that nonreaders of German can get the thrust of Shoda's development. I encourage others to comment and provide alternative readings if they are interested. This posts deals with part I of the eight part development.

One key term I have not translated well (excuse lack of umlauts) is Verknupfung, which can mean short cut or abbreviation. Behaviourally, this looks like a binary operation written in infix notation, and section 5 supports this. There are ways of representing groups as structures with three binary operations (of type $\langle 2,2,2 \rangle$ in more modern terminology), and in section five the observation is made that (the variety of) groups expressed this way is definitionally equivalent to one in which two of the operations are term-defined using the third operation. Also noted are that if just the multiplication operation is used, the equational presentation so restricted only gives semigroups, and if one insists on commutativity for multiplication, one gets abelian groups, So understanding the example in section five helps in reading the paper. However, there is the suggestion earlier that the shortcuts of interest are those that are invariant across a certain partition of the underlying set, and that they may not represent a binary operation so much as something that takes members from a pair of classes in the partition to another class of that partition, so reading Verknupfung as "infix binary operation" may miss something.

Another word with which I grapple is primitiv (feminine form in German is primitive). In section 3 it is applied to class of algebras that appear to me to be equationally defined, and so I think varieties. A further paragraph supports this, mentioning semi groups, groups, rings, lattices as primitive, but fields not so (because of division by zero), and integral domains are also not primitiv, the reason given that a quotient (homomorphic image) of an integral domain may not be an integral domain. However, I am having difficulty understanding the literal translation.

I will edit this later to add more about paper I.

Gerhard "I Am Not A Roman" Paseman, 2017.10.24.

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  • $\begingroup$ How are your efforts going? Cheer up! :-) $\endgroup$
    – Jose Brox
    Commented Nov 16, 2017 at 16:56
  • $\begingroup$ @Jose, they go slow. However, I could use a lot of CPU time on a fast computer for another project. I might speed myself up on translation if I can get a big compute job done. I invite you to email me if you wish to discuss specifics. Gerhard "Quid Pro Quo Pro Processing" Paseman, 2017.11.18. $\endgroup$
    – Gerhard Paseman
    Commented Nov 18, 2017 at 22:51
  • $\begingroup$ (1) In ordinary speech, "Verknüpfung" means "joining" (probably etymologically connected with "Knopf" = "button"), so in mathematics "binary operation" seems right. (2) What you wrote about "invariant across a certain partition" sounds to me like a description of how an operation on a set can (in favorable cases) induce an operation on a quotient set. $\endgroup$
    – Andreas Blass
    Commented May 14, 2020 at 17:11

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