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Definition

Given an a algebraic sequentially limit Cauchy complete Archimedean ordered field $\mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}$ F \mathbb{R}, a algebraic real limit geometric algebra is a $\mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}$ F \mathbb{R}-geometric algebra .$A$ such that the limits preserve the ring structure and the grade projection operation: