Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is simply that

1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside it. But I seem to gather from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question. What do Tamura and Shafer really prove in their paper? Is their proof critically depending on the finiteness of the groups involved? Does their proof also cover the case where at least one of $H$ and $K$ is a group?

Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is simply that

1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside it. But I seem to gather from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question(s). What do Tamura and Shafer really prove in their paper? Is their proof critically depending on the finiteness of the groups involved? Does their proof also cover the case where at least one of $H$ and $K$ is a group?

Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is simply that

1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside it. But I seem to gather from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question. What do Tamura and Shafer really prove in their paper? Is their proof critically depending on the finiteness of the groups involved?

Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is simply that

1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside it. But I seem to gather from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question. What do Tamura and Shafer really prove in their paper? Is their proof critically depending on the finiteness of the groups involved? Does their proof also cover the case where at least one of $H$ and $K$ is a group?

Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is that

1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside the paper. But it seems from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question. What do Tamura and Shafer really prove in their paper?

Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is simply that

1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside it. But I seem to gather from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question. What do Tamura and Shafer really prove in their paper? Is their proof critically depending on the finiteness of the groups involved?