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Embedding semigroups in groups: not as simple as it might seem

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Abstract

We consider the investigation of the embedding of semigroups in groups, a problem which spans the early-twentieth-century development of abstract algebra. Although this is a simple problem to state, it has proved rather harder to solve, and its apparent simplicity caused some of its would-be solvers to go awry. We begin with the analogous problem for rings, as dealt with by Ernst Steinitz, B. L. van der Waerden and Øystein Ore. After disposing of A. K. Sushkevich’s erroneous contribution in this area, we present A. I. Maltsev’s example of a cancellative semigroup which may not be embedded in a group, which showed for the first time that such an embedding is not possible in general. We then look at the various conditions that were derived for such an embedding to take place: the sufficient conditions of Paul Dubreil and others, and the necessary and sufficient conditions obtained by A. I. Maltsev, Vlastimil Pták and Joachim Lambek. We conclude with some comments on the place of this problem within the theory of semigroups, and also within abstract algebra more generally.

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Notes

  1. figure a

    . Besides Russian and Ukrainian, Sushkevich also published papers in English, German and French, in which he used a German transliteration of his name: ‘Suschkewitsch’. Certain of his relevant publications may therefore be found in the bibliography under this version of his name.

  2. A semigroup \(S\) in which either \(ac=bc\) or \(ca=cb\) implies that \(a=b\), for any \(a,b,c\in S\). This does not, of course, guarantee that \(c\) has an inverse element in the traditional group-theoretic sense. If \(S\) has a zero element, then we must make the additional requirement that \(c\) be nonzero.

  3. figure b

    . I choose to omit the silent Cyrillic letter

    figure c

    from my transliteration, although its presence is sometimes indicated (after transliteration into Latin letters) by ’. Note that Maltsev’s name has also been transliterated as ‘Malcev’. Indeed, the 1937 paper that will be of interest to us in Sect. 3.2 was published under this name, and so it appears in this form in the bibliography.

  4. For a ring has no zero divisors if and only if its multiplication is cancellative (in the sense of note 2). Suppose first of all that a ring \(R\) has no zero divisors, and that \(ac=bc\), for \(a,b,c\in R\), with \(c\ne 0\). Then \(ac-bc=(a-b)c=0\), from which it follows that \(a-b=0\), whence \(a=b\). The other part of the cancellation law is similar. Conversely, suppose that \(R\) is cancellative. For \(a,b,c\in R\), with \(c\ne 0\), we take the equality \(ac=bc\) (from which it follows that \(a=b\)) and set \(b=0\). \(\square \)

  5. For a general discussion of fields of fractions, see Cohn (1995, Chap. 1).

  6. ‘Welche kommutativen Ringe besitzen einen Quotientenkörper? Oder, was auf dasselbe hinauskommt, welche lassen sich überhaupt in einen Körper einbetten?’ (van der Waerden 1930, p. 47).

  7. ‘Die Möglichkeit der Einbettung nichtkommutativer Ringe ohne Nullteiler in einen sie umfassenden Körper bildet ein ungelöstes Problem, außer in ganz speziellen Fällen’ (van der Waerden 1930, p. 49).

  8. For biographies of Ore, see Anon (1970) and Aubert (1970).

  9. Ore labelled his condition in this manner because it was the fifth multiplicative condition in his list.

  10. Not to be confused with the notion of a (von Neumann) regular ring (see, for example, Goodearl 1979). What Ore called a regular ring (with identity) is now termed a right Ore domain (Coutinho 2004, p. 258).

  11. This may in fact be stated as a slightly stronger result: a commutative semigroup can be embedded in a group if and only if it is cancellative (see Clifford and Preston 1961, Sect. 1.10).

  12. A two-sided ideal in a semigroup is defined in the same way as for a ring, but without any mention of addition; see Hollings (2009a, p. 502).

  13. Sushkevich is one of only two authors I have found who used the term ‘Semigruppe’ (at least in German), the modern German term for a semigroup being ‘Halbgruppe’. The other author was Fritz Klein-Barmen (1943). Moreover, in the Ukrainian of his 1935 embeddings paper, Sushkevich used the term ‘

    figure g

    ’, in contrast to the modern Ukrainian ‘

    figure h

    ’ (‘

    figure i

    -’ and ‘

    figure j

    -’ both being Ukrainian prefixes denoting ‘half-’ or ‘semi-’). We will see similarly unusual terminology in the work of Vlastimil Pták in Sect. 6 (see, in particular, note 44).

  14. For, suppose that \(S\) is a finite cancellative semigroup. It follows immediately from cancellation that \(sS=S=Ss\), for any \(s\in S\). From this, we conclude that, for a fixed \(s\), there exists \(e\in S\) such that \(se=s\). But \(e\) is in fact a right identity for any element of \(S\), since any \(t\in S\) may be written in the form \(t=us\). It follows further that \(e^2=e\). Consider the product \(es\), for any \(s\in S\). We may write \(s\) as \(eu\), for some \(u\in S\). Then \(es=e^2u=eu=s\). Thus, \(e\) is a two-sided identity for \(S\). As to inverses, it follows from \(sS=S\) that there exists \(s'\in S\) such that \(ss'=e\). Furthermore, \(ss's=es=s=se\), and so it follows from cancellation that \(s's=e\). \(\square \)

  15. Not long after this paper was published, Sushkevich presented the ‘Quotientenbildung’ in both his short Ukrainian textbook Elements of new algebra (

    figure k

    ) (Sushkevich 1937b, Sect. 14) and in the 3rd Russian edition of his Foundations of higher algebra (

    figure l

    ) (Sushkevich 1937a, Sect. 236). In neither case, however, did he mention the semigroup case.

  16. ‘gewiß nicht trivial’.

  17. figure n

    ’ (Sushkevich 1937c, p. 91, footnote 1).

  18. Indeed, this is what is going on in the background of the ‘Quotientenbildung’. For more on this idea, see Howie (1995, Sect. 1.5) or Hollings (2007).

  19. figure s

    .

  20. For biographies of Dubreil, see Lesieur (1994) and Lallement (1995), and also the autobiographical material contained in Dubreil (1981, 1983).

  21. ‘Ultérieurement, cet auteur a donné des conditions nécessaire et suffisantes pour qu’il existe un groupe contenant un semi-group donné’ (Dubreil 1943, p. 626). Note that in his use of the term ‘semi-groupe’ to mean a cancellative semigroup, Dubreil was almost certainly following de Séguier (1904), who had originally coined the term for this notion (see Hollings 2009a, Sect. 3); Dubreil termed a general semigroup a ‘demi-groupe’.

  22. ‘Ce Mémoire et sa traduction m’ont été aimablement communiqués par MM. B. L. van der Waerden et H. Richter, auxquels j’exprime mes sincères remerciements’ (Dubreil 1943, p. 626, footnote 2).

  23. ‘Mais un autre résultat, d’un degré de généralité intermédiaire, et particulièrement intéressant par sa maniabilité et ses possibilités d’application, a été donné dès 1931 par O. Ore ...’ (Dubreil 1943, p. 626).

  24. A detailed account of this paper, as well as a general discussion of the embedding problem, may be found in a series of articles by Faisant (1971a, b, c, 1972).

  25. His papers were published mostly in Norwegian journals, and have thus not been widely available internationally (see Fenstad 1996); we may speculate that the opposite was also true and that Skolem did not have wide-ranging access to foreign sources.

  26. A non-exhaustive list of biographical articles on Maltsev is: Aleksandrov et al. (1968), Anon (1989), Dimitrić (1992), Gainov et al. (1989), Khalezov (1984), Kurosh (1959), Malcev (2010) and Nikolskii (1972). Maltsev also features in Sinai (2003, pp. 559–560).

  27. Although, as noted in the introduction, I use the terms ‘embed/embedding/embeddability’ elsewhere, I nevertheless choose to translate ‘

    figure u

    ’ as ‘immersion’ here, rather than ‘inclusion’ or ‘insertion’, in order to mirror Maltsev’s own English terminology in Malcev (1937). Another often-used Russian word for ‘embedding’/‘imbedding’ is ‘

    figure v

    ’ (‘containment’), but this word only seems to have come into use in this context with later papers (see Schein 1961, for example). Other terms are ‘

    figure w

    ’ (‘immersion’), as used in Lyapin (1960), and ‘

    figure x

    ’ (‘enclosure’), as used in the Russian translation of Clifford and Preston (1961).

  28. figure aa

    ’ (Maltsev 1939, p. 331).

  29. More specifically, the notation used in Bush (1961) is more or less identical to that of Maltsev; the notation used in Bush (1963a) is somewhat simpler. Other accounts of Maltsev’s conditions may be found in Clifford and Preston (1967, Sect. 12.6) and Cohn (1965, Sect. VII.3).

  30. This part of the discussion is drawn from Bush (1961). Bush termed these \(2n\)-tuples \(l\) -sequences; Clifford and Preston (1967, Sect. 12.6) referred to them as Malcev sequences.

  31. It should be noted that this terminology was introduced by Bush, and was not used by Maltsev.

  32. Clifford and Preston (1967, Sect. 12.6) subsequently referred to this as the locked equation; to them, a normal system of equations was a Malcev system.

  33. Maltsev (1939, p. 335); see also Bush (1963a, Theorem 2) and Clifford and Preston (1967, Theorem 12.17).

  34. figure aj

    ’ (Maltsev 1939, p. 336).

  35. figure ak

    ’ (Maltsev 1939, p. 336).

  36. Note that the ‘\(\langle \cdot |\cdot \rangle \)’ notation was not used by Maltsev.

  37. Maltsev (1940, Theorem 4); see also Bush (1961, Theorem 6.2) and Clifford and Preston (1967, Sect. 12.8).

  38. There is a particularly large number of biographical articles on Pták, including: Anon (2000), Fiedler (2000), Fiedler and Müller (2000a, b), Vavřín (1995, 1996a, b) and Vrbová (1985).

  39. The candidate of sciences degree is an Eastern European qualification, equivalent to a Western PhD; it is often followed, as in Pták’s case, by a higher research degree, the doctor of sciences, which is sometimes compared with a DSc, or a German Habilitation.

  40. ‘Tento problém je dåuležitý v algebře samé již proto, že každý okruh bez dělitelåu nuly se stane po odstranění prvku \(0\) semigrupou vzhledem k násobení’ (Pták 1953a, p. 259). In fact, the removal of 0 is not necessary.

  41. ‘Význam podobných výsledkåu pro analysu je nasnadě’ (Pták 1953a, p. 260).

  42. All three papers carry (essentially) the same title, but in different languages: ‘Immersibility of semigroups’ in English, ‘

    figure at

    ’ in Russian, and ‘Vnořitelnost semigrup’ in Czech (for some comments on the terminology used in these titles, see note 44 below). Excluding blank pages, the English version is 14 pages long. The Russian version is 22 pages in length—the extra pages are accounted for by the fact that this version was expanded very slightly (in particular, some comments were added which draw connections with Maltsev’s work), and the fact that the typeface is slightly larger with respect to the page than in the English version. The Russian version also features a further 3 pages of English summary. The Czech version is a slightly modified Czech translation of the English summary from the Russian version. We note also the existence of a 1 page French summary of Pták’s work, based upon the Russian version: Thibault (1953b, p. 10).

  43. My first thought upon seeing the ‘M.’ here was that Pták might have meant ‘Monsieur Maltsev’. However, leaving aside the question of why Pták would have addressed Maltsev in this manner, a glance through the paper suggests that this is not in fact the case: none of the other authors is referred to as ‘Monsieur’—Ore appears as ‘O. Ore’ and Dubreil as ‘P. Dubreil’; van der Waerden is not afforded any initials at all.

  44. Curiously, the Russian version of Pták’s work employs the term ‘

    figure au

    ’ (‘semigruppa’) for semigroup, rather than the usual Russian term ‘

    figure av

    ’ (‘polugruppa’). Perhaps he adopted this term to mirror Dickson’s notion of ‘semigroup’, that is, to refer specifically to a cancellative object—much as Sushkevich appears to have done with his German term ‘Semigruppe’ (see note 13). I have found no other author who uses the term ‘

    figure aw

    ’. Noteworthy terminology also appears in the Czech version of Pták’s work, where he used the term ‘semigrupa’; the modern Czech word for ‘semigroup’ is ‘pologrupa’.

  45. Pták (1949, Theorem 3.6); see also Clifford and Preston (1967, Corollary 12.8). An equivalent version of this result, stated in terms of congruences rather than normal subgroups, may be found in Clifford and Preston (1967, Theorem 12.7).

  46. We note also the existence of what is effectively a French translation of this paper: Thibault (1953b).

  47. These biographical details on Lambek are drawn from Barr (1999) and Barr et al. (2000).

  48. These two are bound together into a single volume in the McGill University Library, with the dissertations labelled as parts A and B, respectively.

  49. Joachim Lambek, private communication, 14th June 2011.

  50. Bush (1961, 1963a). Indeed, Bush’s approach differed considerably in presentation from Lambek’s. As Bush noted:

    We use polyhedra as a convenient model and mnemonic device, but not as part of the proof (Bush 1961, p. 30).

    Thus, in contrast to the situation with Maltsev’s conditions, it is inappropriate for us to follow Bush’s exposition if we are to explore Lambek’s original conditions.

  51. Unlike Maltsev, who referred to the ‘immersibility’ of a semigroup, and spoke of ‘immersing’ it in a group, Lambek used a mixture of terminology: for him, a semigroup that was ‘immersible’ could be ‘embedded’ in a group. I have standardised the terminology as ‘embed’/‘embedding’/‘embeddability’ throughout this section.

  52. Bush (1963a, p. 52). Bush called this a Lambek polyhedral condition but we drop the ‘polyhedral’ not only to avoid confusion with the polyhedral condition, but also for reasons of symmetry with the ‘Maltsev conditions’ of Sect. 5, with which these Lambek conditions will shortly be connected. Lambek himself gave no special name to these individual conditions.

  53. The ‘\(f(C)\)’ used here is Lambek’s notation; Clifford and Preston used ‘\(C\alpha \)’.

  54. Lambek (1951, p. 35); see also Clifford and Preston (1967, Theorem 12.16) and Bush (1963a, Theorem 3).

  55. For the full details of necessity, and for the proof of sufficiency, see Clifford and Preston (1967, Sect. 12.5).

  56. Mathematical Reviews contains reviews of five papers by Bush: the two already cited, a further paper concerning an embedding theorem given by Adyan (Bush 1963b), an article containing proposals for the standardisation of certain terminology relating to functions and binary relations (Bush 1969), and a much later paper on a topic related to computer science, published in a Turkish journal (Bush 1989). A casual online search reveals also a precursor to the cited 1969 article (Bush 1967), an elementary mathematics textbook (Bush and Obreanu 1965), and a problem contributed to The American Mathematical Monthly (Bush 1962; Heuer 1963). Furthermore, the search also yields the information that Bush was elected to ordinary membership of the American Mathematical Society in April 1965 (Green and Pitcher 1965, p. 590).

  57. In fact, Bush proved our Theorem 17 and Corollary 4 in a slightly more specific instance, with ‘Lambek condition’ replaced by ‘Lambek associative condition’, where a ‘Lambek associative condition’ is a special type of Lambek condition, identified by Bush, which gives a pictorial representation of the associative law. This arose from Bush’s observation that, whilst Lambek had proved the necessity of all Lambek conditions for the embeddability of a semigroup in a group, his proof of sufficiency used only those Lambek conditions of a certain form, namely, the Lambek associative conditions. As a result, Bush worked almost exclusively in terms of Lambek associative conditions; in particular, he proved the necessity part of Lambek’s Embeddability Theorem for Lambek associative conditions only. Since Bush’s definition of a Lambek associative condition is rather involved, I have avoided it here. The interested reader is referred to Bush’s original exposition: see Bush (1961, Sect. 3.3) or Bush (1963a, Sect. 2.2).

  58. Unlike Bush, Jackson pursued mathematical research beyond his MA dissertation: Mathematical Reviews lists 16 publications, ranging from 1965 to 1985, most of them on potential theory.

  59. See note 14.

  60. Bush (1961, Sect. 4.3). It is also worth noting that, in spite of our earlier observation that their necessity follows from that of the Lambek conditions, Bush presented Jackson’s proof of the necessity of the lunar conditions (in his Sect. 4.2). This is because of the comment made in note 57 above, concerning the ‘Lambek associative conditions’: Bush did not prove the necessity of arbitrary Lambek conditions, only of the so-called Lambek associative conditions. The lunar conditions are Lambek conditions, but they are not Lambek associative conditions, so their necessity does not follow from the earlier necessity proof in Bush’s presentation. He therefore needed to prove this separately.

  61. The necessity of the dual lunar conditions may be shown in much the same way as for the lunar conditions.

  62. Clifford and Preston stated this theorem in rather more complicated (and precise) terms, but this simplified statement will suffice for our purposes, not least because their notation for Maltsev conditions differs from that used here.

  63. We note that, besides the work on lunar conditions, there have been other attempts to unify the approaches of Maltsev and Lambek. For example, the work of Krstić (1985), which gave a new geometrical interpretation of Maltsev’s conditions, and used this to draw connections with Lambek’s. Johnstone (2008) united the approaches of Maltsev and Lambek in a single categorical method, and then extended this to the embedding of categories in groupoids.

  64. Indeed, the same might also be said of Ore’s adaptation of the ‘Quotientenbildung’ to the non-commutative case.

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Correspondence to Christopher Hollings.

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Communicated by: Jeremy Gray.

This article is an expanded version of Chapter 5 of my book Hollings (2014b) and, as such, contains research carried out at the Mathematical Institute of the University of Oxford with the support of research project grant F/08 772/F from the Leverhulme Trust. I would like to thank Jackie Stedall and Peter M. Neumann for their critical comments on an earlier draft of this paper, and also Joachim Lambek for his permission to quote from private correspondence.

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Hollings, C. Embedding semigroups in groups: not as simple as it might seem. Arch. Hist. Exact Sci. 68, 641–692 (2014). https://doi.org/10.1007/s00407-014-0138-4

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