# What kind of structure is an ISP?

In "Natural duality, Modality, and Coalgebras", in his thesis Meaning and Duality - From Categorical Logic to Quantum Physics, and elsewhere Yoshihiro Maruyama talks about $$\mathrm{ISP}$$, $$\mathrm{ISP}$$(M), how it is composed of $$\mathrm{I}$$, $$\mathrm{S}$$ and $$\mathrm{P}$$, but I can't figure out what these stand for.

It must be a well known concept as it seems to be assumed what it means.

• The first abstract says that the following paper "introduces a new notion of $\mathbb{ISP_{M}}$ (emphasis mine). The second excerpt gives explicitly a definition of an $\mathbb{ISP_{M}}$ (Definition 2.2), and also refers to 3 references to consult for background. Commented Nov 23, 2019 at 6:46
• yes, there are a huge number of references in the thesis. So I was getting a bit overwhelmed. In any case an interesting thesis. I recommend it. :-) I read a book on universal algebra a year and a half ago, but could not remember it. I'll go an consult it now. Commented Nov 23, 2019 at 7:04

$$\mathbb{I}$$, $$\mathbb{S}$$, and $$\mathbb{P}$$ are each operators on classes of structures:

• $$\mathbb{I}(\mathcal{K})$$ is the class of structures isomorphic to a structure in $$\mathcal{K}$$.

• $$\mathbb{S}(\mathcal{K})$$ is the class of substructures of structures of $$\mathcal{K}$$.

• $$\mathbb{P}(\mathcal{K})$$ is the set of all products of structures in $$\mathcal{K}$$.

• Note that when $$\mathcal{K}=\{X\}$$ consists of a single structure, $$\mathbb{P}(\mathcal{K})$$ is the class of powers of that structure.

In particular, $$\mathbb{ISP}(\mathcal{K})$$ is the class of structures isomorphic to a substructure of a product of structures in $$\mathcal{K}$$. A bit of abuse of notation occurs here - an individual structure $$X$$ is conflated with its singleton $$\{X\}$$ and composition is given by juxtaposition, so that e.g. "$$\mathbb{ISP}(L)$$" is written for "$$\mathbb{I(S(P}(\{L\})))$$."

The paper's "$$\mathbb{P}_M$$" is then a modification of the operator $$\mathbb{P}$$, gotten by replacing "power" with "modal power."

A quick aside:

Another operator of this type which is quite important is $$\mathbb{H}$$, taking a class of structures to the class of homomorphic images of structures in the original class. Birkhoff's theorem says that the classes of structures described by equational theories are exactly those which are closed under $$\mathbb{H}$$, $$\mathbb{S}$$, and $$\mathbb{P}$$, and more generally that $$\mathbb{HSP}(\mathcal{K})$$ is always the class of structures satisfying all equations true of every element of $$\mathcal{K}$$.

But here, for whatever reason, $$\mathbb{H}$$ does not seem relevant.