Abstract
Mereotopology is a concept rooted in analytical philosophy. The phase-field concept is based on mathematical physics and finds applications in materials engineering. The two concepts seem to be disjoint at a first glance. While mereotopology qualitatively describes static relations between things, such as x isConnected y (topology) or x isPartOf y (mereology) by first order logic and Boolean algebra, the phase-field concept describes the geometric shape of things and its dynamic evolution by drawing on a scalar field. The geometric shape of any thing is defined by its boundaries to one or more neighboring things. The notion and description of boundaries thus provides a bridge between mereotopology and the phase-field concept. The present article aims to relate phase-field expressions describing boundaries and especially triple junctions to their Boolean counterparts in mereotopology and contact algebra. An introductory overview on mereotopology is followed by an introduction to the phase-field concept already indicating its first relations to mereotopology. Mereotopological axioms and definitions are then discussed in detail from a phase-field perspective. A dedicated section introduces and discusses further notions of the isConnected relation emerging from the phase-field perspective like isSpatiallyConnected, isTemporallyConnected, isPhysicallyConnected, isPathConnected, and wasConnected. Such relations introduce dynamics and thus physics into mereotopology, as transitions from isDisconnected to isPartOf can be described.
Highlights
The term mereology originates from the Ancient Greek word, μερoς + −logy (“study, discussion, science”), while the term topology originates from the Ancient Greek word, τóπoς + −(o)logy (“study of, a branch of knowledge”)
A dedicated chapter—in a summarizing way— compares expressions derived from the phase-field concept with their counterparts in region connect calculus and in classical mereotopology, respectively
Further notions become possible on the basis of the phase-field concept, such as wasPhysicallyConnected; isPathConnected, or isEnergeticallyConnected are shortly introduced and provide a promising outlook on possible future developments of mereotopology towards “mereophysics”
Summary
The term mereology originates from the Ancient Greek word, μερoς (méros, “part”) + −logy (“study, discussion, science”), while the term topology originates from the Ancient Greek word, τóπoς (tópos, “place, locality”) + −(o)logy (“study of, a branch of knowledge”). The Region-Based Theory of Space (RBTS), going back to Whitehead [3] and de Laguna [19], has as primitives the more realistic notion of a region as an abstraction of a finite-sized physical body, together with some basic relations and operations on regions, such as the isConnected or isPartOf relations This is one of the reasons why the extension of mereology, complemented by these new relations, is commonly called mereotopology “MT”. Sequent Algebra replaces the contact between two regions with a binary relation between finite sets of regions and a region satisfying some formal properties of the Tarski consequence relation Another approach to multiple connected regions is, e.g., the mereology for connected structures [38]. The major focus and innovative topics of the present article are (i) a step-by-step comparison of the logical (FOL) concepts of Boolean Algebra as used in mereology, region connect calculus and contact algebra with the algebraic and field theoretic concepts of the phase-field method, (ii) the identification of similarities and discrepancies between these two concepts for each of these steps, and (iii) the identification and preliminary discussion of the shortcomings of either method in a summarizing conclusion
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
