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YCor
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Removed deprecated (abstract-algebra) tag - see the tag info: https://mathoverflow.net/tags/abstract-algebra/info (if there are some other suitable tags, choose them instead.)
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Martin Sleziak
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Adding more links to references on request
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Jose Brox
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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

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.

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.

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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Jose Brox
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