Let $S$ be a (multiplicatively written) semigroup. Equipped with the (binary) operation of setwise multiplication $(X, Y) \mapsto \{xy \colon x \in X, \, y \in Y\}$, the family of all non-empty subsets of $S$ is then itself a (multiplicatively written) semigroup, herein denoted by $\mathcal P(S)$ and called the power semigroup of $S$.
To the best of my knowledge, power semigroups were first explicitly studied by Tamura and Shafer in the late 1960s. In particular, the earliest reference I've been able to track down is
T. Tamura and J. Shafer, On power semigroups, Math. Jap. 12 (1967), 25-32
Unfortunately, I don't have a copy of the paper and my understanding of its content is limited to a zbMATH review by McAlister. In any case, a question arisen (I believe) from Tamura and Shafer's work was to prove (or disprove) that $\mathcal P(S)$ is (semigroup-)isomorphic to the power semigroup $\mathcal P(T)$ of a semigroup $T$ (if and) only if $S$ is isomorphic to $T$. The question was answered (in the negative) by E. M. Mogiljanskaja, see
Non-isomorphic semigroups with isomorphic semigroups of subsets, Semigroup Forum 6 (1973), 330-333
and references therein. But Mogiljanskaja writes,
The problem was proposed by: B. M. Schein (1960), T. Tamura (1967) [5] and others.
Here, [5] is
T. Tamura, Unsolved problems on semigroups, Sem. Reports. of Math. Sci., (1967), 33-35.
Unfortunately, I don't have a copy of this last paper either. So, I'm writing to ask if anybody can provide additional details and shed light on (some of) the unclear aspects of this story. In particular, I've the following questions:
Q1 (kind of answered in the "Edit" below). Is it really that power semigroups were first explicitly considered by Tamura and Shafer? I've tried to look up the keywords "power" and "global" in Howie's and Clifford & Preston's monographs on semigroups, but haven't come up with anything (some people refer to power semigroups as globals). Q2. Which work of Schein is Mogiljanskaja referring to in the excerpt from their 1973 paper that I quoted in the above? This may be relevant to Q1, as Mogiljanskaja's words seem to suggest that Schein (I suppose this is Boris Moiseyevich Schein) had already considered power semigroups as early as 1960.
Edit (a couple of hours later). "Systematically" may be a more appropriate choice of words than "explicitly". Indeed, I've just learned from Sect. 2 of
C. Brink, Power structures, Algebra Universalis, 30 (1993), 177-216
that
According to Birkhoff [1948] the concept of power algebra originated with Frobenius, in the context of group theory. Any subset of a group was referred to as a complex [...]. Some discussion of [...] complexes can often be found in the older introductory group theory textbooks, e.g. Hall [1959] and MacDonald [1968]. Power algebras are therefore also sometimes known as complex algebras - for example, they are so referred to in the brief mention they get in Grätzer [1979].
Brink's paper was brought to me by Paolo Lipparini's response to a related question on this forum (the "Related" column on the right has proven useful once again!). On the other hand, one can read from p. 5 of
H.B. Hamilton and T.E. Nordahl, Tribute for Takayuki Tamura on his 90th birthday, Semigroup Forum, 79 (2009), 2-14
that "Tamura [...] was an initiator of the study of power semigroups of a semigroup" (emphasis mine). This seems to answer my Q1 and supports the idea that, while power algebras had been considered and formally introduced earlier, the explicit and systematic study of power semigroups started with Tamura's work in the late 1960s.
