Hereditary sigma-ring generated by

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Definition

Given a system of sets, [ilmath]S[/ilmath], the hereditary [ilmath]\sigma[/ilmath]-ring generated by [ilmath]S[/ilmath] is^[1]:

The smallest hereditary [ilmath]\sigma[/ilmath]-ring containing [ilmath]S[/ilmath]

We denote this [ilmath]\mathcal{H}_{\sigma R}(S)[/ilmath]^{[Note 1]}

[ilmath]\mathcal{H}(\sigma_R(S))=\sigma_R(\mathcal{H}(S))[/ilmath] - see Notes:Hereditary sigma-ring

↑ If [ilmath]\mathcal{H}(A)[/ilmath] denotes the hereditary system of sets generated by [ilmath]A[/ilmath] and [ilmath]\sigma_R(B)[/ilmath] the sigma-ring generated by [ilmath]B[/ilmath], then we have [ilmath]\mathcal{H}(\sigma_R(A))=\sigma_R(\mathcal{H}(A))[/ilmath]