Algebraic Structures & Order Relations

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Abstractions that define algebraic and order-theoretic structures — group and ring constructions (direct product of groups, nilpotent algebra, metanilpotent group, primal ideal), order-theoretic notions (atom, linear order, product order), and structural classification devices borrowed into other fields such as binade and set theory in music.

18 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Acceptable ring — In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.
• Algebraic Structure — One or more carrier sets equipped with typed operations, distinguished elements, and laws that define an algebraic kind and its structure-preserving mappings.
• Atom (Order Theory) — In the mathematical field of order theory, an element a of a partially ordered set with least element 0 is an atom if 0 0 has an atom a below it, that is, there is some a such that b ≥ a :> 0.
• Binade — In software engineering and numerical analysis, a binade is a set of numbers in a binary floating-point format that all have the same sign and exponent.
• Direct product of groups — In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted .
• Double Mersenne number — In mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2p-1}-1 where p is prime.
• Linear order — In mathematics, a total order or linear order is a partial order in which any two elements are comparable.
• Metanilpotent Group — In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent.
• Multiset Abstract Data Type — In computer science, a set is an abstract data type that can store distinct values, without any particular order.
• Nilpotent algebra — In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero.
• Primal ideal — A proper ideal I of a commutative ring A is said to be primal if the elements that are not prime to it form an ideal.
• Product order — In mathematics, given partial orders \preceq and \sqsubseteq on sets A and B , respectively, the product order (also called the coordinatewise order or componentwise order ) is a partial order \leq on the Cartesian product A \times B.
• Profunctor — In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.
• Quotient Algebra — An algebra of congruence classes whose operations descend from a given algebra independently of representative choice.
• Semigroup with Involution — An associative algebraic system with a self-undoing unary operation that reverses product order.
• Set theory (music) — The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.
• Stoneham number — In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G.
• Zero Divisor — In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective.