Íslem

Íslem is a fictional mathematical concept used in thought experiments and educational puzzles to illustrate the properties of binary operations that need not be associative. The term originates in a constructed language employed by puzzle communities, where íslem is offered as an abstract operation rather than a defined real-world process.

Definition and scope: An íslem is a binary operation ⊙ defined on a set S such that for all a, b in S, a ⊙ b is in S. Unlike many standard operations, an íslem is not required to be commutative or associative. Some instances may have an identity element, others may not. The family is intentionally broad to allow exploration of how different rules affect structure.

Example: Let S = {0,1} and define a ⊙ b = a AND NOT b. This operation is closed on S, non-commutative (0 ⊙ 1 = 0, 1 ⊙ 0 = 1), and non-associative (for a = 1, b = 0, c = 1, (a ⊙ b) ⊙ c = 0 whereas a ⊙ (b ⊙ c) = 1).

Usage and significance: Íslems are used in teaching to demonstrate non-associativity in small finite sets, and in recreational puzzles to create nontrivial binary-operation tables. They also serve as a flexible template for comparing how different rule choices shape algebraic behavior.

Relation to established concepts: Although fictional, íslems share features with non-associative algebras and quasigroups, which study binary operations with varying degrees of invertibility and associativity. Descriptions of íslems commonly appear in online forums and speculative mathematics discussions as a didactic tool rather than a formal branch of mathematics.

See also: Non-associative algebra, quasigroup, binary operation, algebraic structure.