What is a word to describe something which cannot be broken down any further?

Question

I don't want to use the word elemental, nor axiomatic, as both terms are already defined and in popular use.

Anyway, the way I want to use the term is not to describe the part, or type of part itself, but the quality that the part possesses; 'Something with which cannot be broken down any further possesses the quality x', where x is the word I seek, in the same way an axiom has the quality of being axiomatic.

sw2015 · Accepted Answer

I think that the word atomic, which is often otherwise used in a physical sense, should be used here.

atomic a·tom·ic /əˈtämik/ adjective

of or relating to an atom or atoms. "the atomic nucleus"

Chemistry (of a substance) consisting of uncombined atoms rather than molecules. "atomic hydrogen"

of or forming a single irreducible unit or component in a larger system.

–Oxford Dictionaries

Tim Scroggins · Answer

"Simplex" is another one that would work.

"Simplex" derives from the same root as "simple," but is used more exclusively in the technical sense of being irreducible in form. I think "simplicity" is the noun form of both. It can be used to mean, "Freedom from division into parts."

Stan · Answer

I try to get to the root of the problem. The root of any problem exists as the basis for all resulting and related things and issues.

The root of a word is the basis of its evolution and usage in our language.

Mathematically, a root is a component of an amount, part of a denominator, a multiplicand.

The root is elemental

WhatRoughBeast · Answer

I'd go for irreducible, "not reducible; incapable of being reduced or of being diminished or simplified further".

alwayslearning · Answer

While I agree that atomic may be the best fit, indecomposable works too.

M-W:

indecomposable
  adjective
  
  :  not capable of being separated into component parts or elements

Franklin · Answer

Ineffable:

that cannot be expressed or described in language. (OED)

What is a word to describe something which cannot be broken down any further?

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