The concept of volumetric spaces has evolved in the Architecture, Engineering, and Construction (AEC) sector, ranging from traditional onsite built spaces to modern modular houses and portable architecture. Despite this diversity, there lacks a comprehensive framework to analyze volumetric space products across dimensions. This paper presents a conceptual framework based on abductive reasoning and qualitative analysis, aiming to explore interdependencies among dimensions. It hypothesizes that volumetric space dimensions are perceived differently and demonstrate interdependencies.

In this paper, we initiate the study of fixed point properties of amenable or reversible semitopological semigroups in modular spaces. Takahashi’s fixed point theorem for amenable semigroups of nonexpansive mappings, and T. Mitchell’s fixed point theorem for reversible semigroups of nonexpansive mappings in Banach spaces are extended to the setting of modular spaces. Among other things, we also generalize another classical result due to Mitchell characterizing the left amenability property of the space of left uniformly continuous functions on semitopological semigroups by introducing the notion of a semi-modular space as a generalization of the concept of a locally convex space.

By now the literature concerning the representation of disjointly additive functionals and operators is quite extensive. A few entries on the subject are [6, 7, 8, 11, 20, 21]. In [7, 8, 17] further references can be found, in [7] the “prehistory” of the subject is also discussed.

To quote a typical result, we may take a 1967 theorem of Drewnowski and Orlicz ([6] Th. 3.2, [17] 12.4) which asserts that, under proper assumptions, an abstract modular (= disjointly countably additive functional) p on a “substantial“ subspace D of L° can be realized by the formula .