On the notion of sub-hyperfield

Question

I am working on some generalization of the paper of Gel'fand, Rybnikov and Stone "Projective orientations of matroids" to the wide context of matroids over hyperfields.

I would like to now if the following definition of sub-hyperfield is correct. Indeed, it seems that in this context there are problems in defining the right notion of "injective" homomorphism.

The definition I need (and I hope that is correct) is the following:

Definition

A sub-hyperfield $\mathbb{H}_{1}$ of a hyperfield $\mathbb{H}_{2}$ is a subset $\mathbb{H}_{1}\subseteq\mathbb{H}_{2}$ that itself is a hyperfield with the operations induced by $\mathbb{H}_{2}$

Clearly, with this definition the inclusion map $i:\mathbb{H}_{1}\longrightarrow\mathbb{H}_{2}$ is an injective hyperfield homomorphism. However, I do not know if the converse is true, that is, if the following statement holds:

Problem

Let $f:\mathbb{H}_{1}\longrightarrow\mathbb{H}_{2}$ be an injective hyperfield homomorphism. Thus, the image $f(\mathbb{H}_{1})\subseteq\mathbb{H}_{2}$ is a sub-hyperfield of $\mathbb{H}_{2}$.

Answer 1

There are two possible notions for a sub-hyperring that make sense:

1. A subset $S$ containing zero and such that for any $x, y\in S$, $x + y \in S$ and $xy \subseteq S$.

2. A subset $S$ such that with the addition $x +' y = (x + y) \cap S$ is a hyperring.

So which notion of sub-hyperring are people using? In Jaiung Jun's paper Algebraic Geometry Over Hyperrings (arXiv) we have

Definition 2.4 Let $R$ be a hyperring. By a hyperring extension of $R$ we mean a hyperring $L$ such that there is an injective homomorphism $i : R \longrightarrow L$ of hyperrings. A sub-hyperring $H$ of $R$ is a subset of $R$ such that $H$ itself is a hyperring with the induced addition and multiplication.

Jun defines homomorphisms to satisfy $\varphi(x + y) \subseteq \varphi(x) + \varphi(y)$ and a strict homomorphism satisfies $\varphi(x + y) = \varphi(x) + \varphi(y)$. Thus Definition 2.4 gives you the second definition (2.) above. We could likewise say that the first definition (1.) defines a strict sub-hyperring.

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For the non-strict definition, we have the Krasner hyperfield $\mathbf{K}$ as a sub-hyperfield of the tropical hyperfield $\mathbf{TR}$ since

$$ \varphi(1_\mathbf{K} + 1_\mathbf{K}) = \{0,\infty\} \subseteq [0,\infty] = \varphi(1_\mathbf{K}) + \varphi(1_\mathbf{K}). $$

But this isn't a strict sub-hyperfield.

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From what I've seen so far working with hyper-structures, I've found it helpful to think about both the strict case and the non-strict case.