The Triomathematics, Trithmetic, Trichometry, Triangulus & Ternary Algebra Expressions that Define the Triophysics and Triological Sciences Related to the Triscalar Mitigation of Non-Ionized Radiation via Bioharmonic Biobalancing Healthy Smart Solar Nanob

The narrative in this paper provides a detailed and descriptive mathematical model for operative Healthy Smart Solar Nanobiotechnology ™ © (Osler, 2021) designed to biobalance a non-ionized irradiated environment. The in-depth mathematics mentioned in this paper are the foundational elements for groundbreaking innovative terminologies and new pioneering scientific fields covered in detail to explicitly explain how Healthy Smart Solar Nanobiotechnology ™ © interacts with and works within the environment. These revolutionary fields of study are directly associated with the mitigating mathematical operation of Solar Smart Nanobiotechnology ™ ©. The mathematics defined in the paper also aid in the creation of new, novel, and original areas and arenas of knowledge that are designed to most accurately describe the trichotomous holistic solutions that can be used to create an all around healthy and safe environment for the well-being of all.

A NOTE ON HILBERT TERNARY ALGEBRAS

We show that every simple abelian real Hilbert ternary algebra is isomorphic to the algebra of Hilbert-Schmidt operators between two real, complex or quaternionic Hilbert spaces, up to a positive multiple of the inner product.

INVOLUTED SEMILATTICES AND UNCERTAINTY IN TERNARY ALGEBRAS

An involuted semilattice <S,∨,-> is a semilattice <S,∨> with an involution-: S→S, i.e., <S,∨,-> satisfies [Formula: see text], and [Formula: see text]. In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice <S,∨,-,1> with greatest element 1 is said to be complemented if it satisfies a∨ā=1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra <T,+,*,-,0,ϕ,1> is a de Morgan algebra with a third constant ϕ satisfying [Formula: see text], and (a+ā)+ϕ=a+ā. If we define a third binary operation ∨ on T as a∨b=a*b+(a+b)*ϕ, then <T,∨,-,ϕ> is a complemented semilattice.

FREE TERNARY ALGEBRAS

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

A Characterization of Finite Ternary Algebras

A ternary algebra is a De Morgan algebra (that is, a distributive lattice with 0 and 1 and a complement operation that satisfies De Morgan's laws) with an additional constant Φ satisfying [Formula: see text], [Formula: see text], and [Formula: see text]. We provide a characterization of finite ternary algebras in terms of "subset-pair algebras," whose elements are pairs (X, Y) of subsets of a given base set ℰ, which have the property X ∪ Y = ℰ, and whose operations are based on common set operations.