Definition:R-Algebraic Structure Monomorphism
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Definition
Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures.
Then $\phi: S \to T$ is an $R$-algebraic structure monomorphism if and only if:
- $(1): \quad \phi$ is an injection
- $(2): \quad \forall k: k \in \left[{1 \,.\,.\, n}\right]: \forall x, y \in S: \phi \left({x \ast_k y}\right) = \phi \left({x}\right) \odot_k \phi \left({y}\right)$
- $(3): \quad \forall x \in S: \forall \lambda \in R: \phi \left({\lambda \circ x}\right) = \lambda \otimes \phi \left({x}\right)$.
That is, if and only if:
- $(1): \quad \phi$ is an injection
- $(2): \quad \phi$ is an $R$-algebraic structure homomorphism.
This definition continues to apply when $S$ and $T$ are modules, and also when they are vector spaces.
Vector Space Monomorphism
Let $V$ and $W$ be $K$-vector spaces.
Then $\phi: V \to W$ is a vector space monomorphism if and only if:
- $(1): \quad \phi$ is an injection
- $(2): \quad \forall \mathbf x, \mathbf y \in V: \phi \left({\mathbf x + \mathbf y}\right) = \phi \left({\mathbf x}\right) + \phi \left({\mathbf y}\right)$
- $(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \phi \left({\lambda \mathbf x}\right) = \lambda \phi \left({\mathbf x}\right)$
Also see
Linguistic Note
The word monomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix mono- meaning single.
Thus monomorphism means single (similar) structure.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 28$