Wonderful! Do you know of any reference to (the 2nd part of) your proof (other than the link to this answer)?

To clarify the previous comment, I'm saying that the case of an algebraically closed field $\mathbb K$ of finite characteristic, though relevant, is trivial, once noticed that every $n$-by-$n$ matrix with entries in $\mathbb K$ has a Jordan normal form. The interesting case is that of a field of zero characteristic (I guess that I should have phrased the OP in a different form).

The p-th power of any n-by-n invertible Jordan block with entries in a field of characteristic p is obviously the identity of the relevant matrix ring. But this counts just as a special (and simple) case, doesn't it? And I'm not interested, as I said, in special cases.

I don't think that your question fits the scope of MO. MSE (math.stackexchange.com) would have been a better choice.

Thanks, I will try to understand as much as I can (hopefully with the help of my German office mate).

Thank you for the references! I'm very interested, but unluckily for me, I cannot read German. So let me ask: Is there any hope that the material of the three papers of the series Banach-Semikategorien has been integrally included in the books mentioned in your answer? In other words, is there anything relevant in the papers that cannot be found in the books?

@Benjamin. I clearly recall that you expressed this point of view in a previous discussion (see mathoverflow.net/questions/106898/references-for-semicategor‌​ies). However, when I say that "grps are to grpds as monoids are to cats, and monoids are to cats as sgrps are to (Mitchell's) semicats", I'm not referring to the choice of terms, but to the meaning of these terms.

@Gerhard. No worries! Instead, thank you for sharing your reflections on this.

(...) This is, of course, unquestionable, and it reflects Andreas & Gerhard's position. Yet, what I'm barely trying to say is that, if we put applications to the side for one moment, we face the fact that there's a higher rationale providing an abstract notion of subobject, which is telling us that, from a conceptual point of view, Grätzer's notion of relative subalg should be privileged, & I'd like to find, if possible, a similar motivation for the notion of partial sgrp. But again, this is just what it is, namely abstract nonsense, and it has little to do here with the everyday practice.

For the record, Evseev's notion of subgrpd is essentially an instance of the more general notion of (partial) subalgebra given by Grätzer at p. 80 of his Universal Algebra (2nd ed., 2008). The notion implied by the categorial point of view is, instead, the one of relative subalgebra, which Grätzer himself introduces a few lines after. To quote him, "In many papers, the authors select one of each [...] and give the reasons for their choice. In the author's opinion, all these concepts have their meirts and drawbacks, and each particular situation determines which one should be used." (...)

Yes, but now that I read myself, I must say that it's expressed in a wrong way. What I mean is that either (xy)z and x(yz) are both defined (and then they're equal to each other), one, and only one, of them is defined, or none of them is defined. In the first two cases, we have a unique unambiguous way to interpret the expression xyz. I'll delete my previous comment to avoid confusion.

You're obviously right, I should have said "conjunctive combination". I insisted on using the adjective "disjunctive" disregarding the 2nd half of the logical implication (i.e., just looking at the 1st operand "Either of (i) ..., or (ii) ..., or (iii) ..."), which doesn't, however, make any sense. Sorry!

I see your point, and I respect it, although I continue to disagree: There's a dose of philosophy in my questions, there is a greater dose of philosophy in mathematics, and it seems that there's even some room for philosophical musings on MO (see, e.g., mathoverflow.net/questions/19644/… or mathoverflow.net/questions/2748/…). Let it be as it may, the discussion so far suggests that my questions have a negative answer, which already counts as a positive answer. P.S.: thanks for the hints!

(...) associativity as the (inclusive) disjunction of three conditions: He says that a partial map $\cdot:\Lambda\times\Lambda\rightharpoonup\Lambda$ is associative if "[...] either of (i) ..., or (ii) ..., or (iii) ... [...]" holds. Now, if my English is not worse than I want to admit to myself, "if either of (i) ..., or (ii) ..., or (iii) ... holds" should mean "if at least one among (i) ..., (ii) ..., or (iii) ... holds". Right?!

But I've never meant that the two definitions of grpd given by Wiki.en are not equivalent to each other. Instead, I said that I don't know which definition (of grpd) you are thinking of when you claim that Exel's sgrpds generalize sgrps in the same way as grpds do with grps: Some people, for instance, use the term "grpd" to refer to Bourbaki's magmas (e.g., J. Howie). Secondly, I'd rather say that grps are to grpds as monoids are to cats, and monoids are to cats as sgrps are to (Mitchell's) semicats, and not to Exel's sgrpds. As for the rest, Definition 2.1 in Exel's paper expresses (...)

@Gerhard. I followed your suggestion, and checked it out for myself, but there's no occurrence of the strings "sociativ" or "semigroup" in the 2nd chapter of the 2008 edition of Grätzer's book (which is apparently the only chapter in the whole text devoted to partial algebras).

(...) on it, and if the questioner is not expressing an interest in any particular application, then I have no doubt as to the answer which I should provide.