What is a "cusp" ("кусок") in relation to Guba's embedding theorem?

Question

I'm confused by the definition of a "cusp" as found in

V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link).

In the words of Mark Sapir (from an answer that has meanwhile been removed from this thread), "cusp" is a «weird translation» into English of the Russian term "кусок" (piece) used by Guba in the original version of his paper. However, this is not the (main) point here: The point is rather that I have seen the same definition repeated elsewhere (with the terms "$s$-piece" or "$S$-piece" used in place of Guba's "кусок"), but continue to find it "strange". Let me try and explain my problem with Guba's definition, before asking a question.

Suppose that $S = \langle X \mid R \rangle$ is a semigroup presentation, and denote by $\mathscr F(X)$ the free monoid over $X$ and by $\ast$ the operation of word concatenation in $\mathscr F(X)$. Following Guba, it seems that

• a defining word (relative to $S$) is, as usual, any word in the set $\bigcup_{(\mathfrak a, \mathfrak b) \in R} \{\mathfrak a, \mathfrak b\}$;

• a cusp (кусок) is a word $\mathfrak u \in \mathscr F(X)$ such that there exist $\mathfrak p, \mathfrak p^\prime, \mathfrak q, \mathfrak q^\prime \in \mathscr F(X)$, with $\mathfrak p \ne \mathfrak q$ or $\mathfrak p^\prime \ne \mathfrak q^\prime$, such that $ \mathfrak p \ast \mathfrak u \ast \mathfrak q$ and $\mathfrak p^\prime \ast \mathfrak u \ast \mathfrak q^\prime$ are defining words.

But this doesn't look very natural. For one thing, what would prevent us from taking $\mathfrak p = \mathfrak p^\prime$ and $\mathfrak q = \mathfrak q^\prime$? Maybe I'm misunderstanding something and what Guba really means is that

• a cusp (кусок) is a word $\mathfrak u \in \mathscr F(X)$ such that there exist $\mathfrak p, \mathfrak p^\prime, \mathfrak q, \mathfrak q^\prime \in \mathscr F(X)$, with $\mathfrak p \ne \mathfrak q$ or $\mathfrak p^\prime \ne \mathfrak q^\prime$, such $ (\mathfrak p \ast \mathfrak u \ast \mathfrak q, \mathfrak p^\prime \ast \mathfrak u \ast \mathfrak q^\prime) \in R$;

or maybe there is a typo in the paper. In fact, I checked the original version of Guba's article, and at least for what concerns the definition of a "cusp" ("кусок"), there hasn't been any typo introduced in the translation process. I've also tried to follow the proof of Theorem 1 in Guba's paper, but there are some details I'm still trying to demystify. On the other hand, it appears that Guba's notion of "cusp" ("кусок") is borrowed from E.V. Kashintsev's paper

Small cancellation conditions and embeddability of semigroups in groups, Int. J. Alg. Comp. 2 (1992), No. 4, 433-441;

except that Kashintsev uses the term "$s$-piece" instead of "cusp" ("кусок") and lets an $s$-piece be a word $\mathfrak u \in \mathscr F(X)$ for which there exist $\mathfrak p, \mathfrak p^\prime, \mathfrak q, \mathfrak q^\prime \in \mathscr F(X)$, with $\mathfrak p \ne \mathfrak p^\prime$ or $\mathfrak q \ne \mathfrak q^\prime$, such that $ \mathfrak p \ast \mathfrak u \ast \mathfrak q$ and $\mathfrak p^\prime \ast \mathfrak u \ast \mathfrak q^\prime$ are defining words (as remarked by Kashintsev, these two words need not be distinct). This is one reason making me think that there is a typo in Guba's definition. The following excerpt if from the English translation of Guba's paper:

The classes $K_p^q$ were studied in [1]. By definition, a semigroup $S$ belongs to the class $K_p^q$ if it can be specified by a corepresentation (1) that satisfies the small cancellation conditions $C_s(p)$ and $D(q)$.

Here, item [1] in the bibliography of Guba's paper is Kashintsev's paper (the same one mentioned in the above) and a semigroup is of class $K_p^q$ if it is isomorphic to a semigroup presentation $S$ with finitely many generators such that (i) no defining word can be expressed as the concatenation of less than $p$ "cusps" and (ii) the left and the right graphs of $S$ (that is, some undirected multigraphs naturally associated with the presentation) have both girth $\ge q$. Then, Guba's embedding theorem (that is, Theorem 1 in Guba's paper) is the statement that every semigroup of class $K_3^2$ embeds into a group. So I would expect Guba's notion of "cusp" ("кусок") to coincide with Kashintsev's notion of "$s$-piece", and yet the two definitions are different (and probably non-equivalent). With all this in mind, I'd like to ask the following:

Question. Can anyone familiar with these things clarify the situation or suggest a reference where Guba's embedding theorem or generalizations of it are discussed as part of a more systematic treatment of the subject (e.g., a chapter in a book concerned with the embedding of a semigroup into a group)?

Answer 1

Update: I had an email exchange with Victor Guba. He has kindly confirmed that there is indeed a typo in (the Russian and English versions of) his paper: a "кусок" (as per his paper) and an "$s$-piece" (as per Kashintsev's paper) are meant to be one and the same thing.

The part below was written before hearing from Guba.

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(A recent comment of user TT_ has reminded me of this question, which has also entered a private exchange with Laura Cossu these days. So I've thought to post an answer in the hope that it can also be useful to someone else or those more familiar than me with the subject can shed further light on the story.)

As far as I can tell, there are two possible interpretations of the facts described in the OP:

• It is implicitly understood in Guba's paper (as suggested by TT_ in their comment under the OP) that either the defining words $w = \mathfrak p \ast \mathfrak u \ast \mathfrak q$ and $w' = \mathfrak p' \ast \mathfrak u \ast \mathfrak q'$ are distinct, or $\mathfrak u$ occurs twice as a subword of $w$.

• There is a typo in Guba's paper (and the typo has unfortunately spread through the literature).

In the first case, a cusp (as per Guba's paper) is always an $s$-piece (as per Kashintsev's paper); and in both cases, an $s$-piece need not be a cusp. E.g., consider the monoid presentation $H := \langle X \mid R \rangle$, where $X$ is the $3$-element set $\{a, b, c\}$ and the only defining relation in $R$ is the pair $(a \ast b \ast a, c \ast b \ast c)$: The only $s$-pieces of $H$ are the empty $X$-word $\varepsilon_X$ and the generators $a$, $b$, and $c$; while, in any case, the only cusps are $\varepsilon_X$, $a$, and $c$ (in particular, $b$ is not a cusp because it only appears in defining words of the form $\mathfrak z \ast b \ast \mathfrak z$).