This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the process of "completing" the binary operation.

Here's the particular situation I'm interested in. Suppose $C$ is a set, and $\cdot$ is a partially defined binary operation on $C$ that is associative in the following sense: for $a, b, c \in C$, if $ab$ and $bc$ are defined, then $(ab)c$ and $a(bc)$ are defined and equal.

Completing the binary operation is essentially the same as "embedding" $C$ into a semigroup. This can be done as follows.

Let $\overline C = C \cup \{\infty\}$ where $\infty$ is not a member of $C$. Extend the binary operation by:

  • If $ab$ is not defined in $C$, then $ab = \infty$.
  • $a\infty = \infty$.
  • $\infty a = \infty$.
  • $\infty\infty = \infty$.

$\overline C$ is the semigroup that I am interested in. Since this construction seems quite natural, I would like to ask if it has a name, and I would really appreciate if you could recommend books that discuss this construction and related ones.

Here's one example of why I'm interested in this construction:

If $C$ is a category with a zero object, the collection of zero morphisms plus $\infty$ would be an ideal in $\overline C$. (Here, members of $C$ are morphisms, not objects. Objects are identified with identity morphisms.)

Obviously, one could define ideals in $C$ without mentioning $\overline C$, but it is a little more cumbersome:

A subset $I$ of $C$ is an ideal of $C$ if for $c \in C$ and $i \in I$,

  • If $ci$ is defined, then $ci \in I$.
  • If $ic$ is defined, then $ic \in I$.

With this definition, we can also say that the collection of zero morphisms in $C$ is an ideal.

However, I feel that this definition of ideal is quite cumbersome because I have to put the condition "if the operation is defined" in every sentence.

  • $\begingroup$ Clearly, it would be cumbersome to write "If $x.y$ is defined". But you can introduce some sort of notation for the domain of your partial operation. I use $x\perp y$ in my papers. Does not look too horrible, IMHO: $$c\perp i\implies ci\in I$$. $\endgroup$ Oct 8 '13 at 19:10
  • $\begingroup$ link.springer.com/article/10.1007%2F978-3-0348-0405-9_5/… Sorry it's only a preview. "Partial" groupoids are also called in literature halfgroupoid. $\endgroup$ May 17 '14 at 7:10

There are several articles of Lyapin on the topic. Here are some of them (they were translated into English):

E. S. Lyapin, The possibility of semigroup continuation of a partial groupoid. Izv. Vyssh. Uchebn. Zaved. Mat., 1989, no. 12, 68–70.

E. S. Lyapin, Partial groupoids that can be obtained from semigroups by restrictions and homomorphisms. Izv. Vyssh. Uchebn. Zaved. Mat., 1989, no. 10, 30–36.

E. S. Lyapin, Internal extension of partial actions to complete associative ones Izv. Vyssh. Uchebn. Zaved. Mat., 1982, no. 7, 40–44.

  • $\begingroup$ Any chance of finding them online? Maybe in the original Russian version? $\endgroup$
    – magma
    Oct 6 '13 at 23:36
  • $\begingroup$ Thank you. I have not been able to find an English version online. Guess I'll have to hit the library to see if it has any. P.S. So I guess the construction here is not as common as I thought it would be :( $\endgroup$
    – Tunococ
    Oct 7 '13 at 0:53
  • $\begingroup$ @Tunococ: I have his paper "Partielle Operationen in der Theorie der Halbgruppen" in: Semigroups, Proceedings of a Conference Held at Oberwolfach, Germany December 16-21,1978. If it is of interest for you, write me by e-mail. $\endgroup$ Oct 7 '13 at 7:12

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