Mathematical Bases for Formulation of Biostratigraphic Methods

SHIONO Kiyoji, YAMAGUCHI Kumiko

doi:10.6010/geoinformatics1990.8.4_227

We propose a mathematical framework for formulation of biostratigraphic methods based on two order relations ≤^__??_ and <<^_??_ on a set whose members are intervals of a totally ordered set (A, ≤) defined by: [a, b] ≤^_??_ [a′, b′] iffa≤a′and b≤b′, and [a, b] <<^_??_ [a′, b′] iffb≤a′or [a, b] = [a′, b′] .Let Σ₀ be a set whose members are individuals of animals and plants which lived within an intervalT, and S₀be a partition ofΣ₀. An element ofS₀is called here“an individual class”.We define a range of an individual class σ, denoted by τ (σ), by union of life intervals of all individuals in σ. This framework enables us to formulate easily several basic concepts. A combination of individual classes which lived together at a timetis given bys (t) = {α|t∈τ (α) } .A range in which the same combination of individual classes lived together is expressed by an equivalence class relative to an equivalence relation-such thatp-qiffs (p) =s (q) .It is possible to order individual classes based on a relation κ such that α κ β iff τ (α) ≤^_??_τ (β) . Further we discuss the details of time classification based on a setSwhich satisfies three assumptions (1) Sis a subset ofS₀such that τ (σ) is an interval ofTfor all σ∈S, (2) (S, κ) is a total ordering, and (3) s (t) is a non-empty set for allt∈T. The results suggest that the proposed mathematical framework is useful to formulate basic concepts of biostratigraphy.

Mathematical Bases for Formulation of Biostratigraphic Methods

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