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Modes and modals. (English) Zbl 0937.08006

Let \(v:F\to N\) be the arity-function of the algebra \((A,F)\). The algebra \((A,F)\) is said to be entropic (or abelian by A. G. Kurosh) if it satisfies the hyperidentity \[ X(Y(x_{11},\dots, x_{1n}),\dots, Y(x_{m1},\dots, x_{mn}))= Y(X(x_{11},\dots, x_{m1}),\dots, X(x_{1n},\dots, x_{mn})) \] for all \(m,n\in v(F)\).
The algebra \((A,F)\) is said to be idempotent if each singleton subset of \(A\) is actually a subalgebra. In other words, the hyperidentity \[ X\underbrace {(x,\dots,x)}_m =x \] is satisfied for each \(m\in v(F)\).
A mode is an idempotent and entropic algebra. A modal is an algebra \((A,+,F)\) such that
(a) \(A(+)\) is a (join) semilattice;
(b) \((A,F)\) is a mode;
(c) \((A,F)\) distributes over \(A(+)\).
The paper gives a self-contained introduction to modes and modals.

MSC:

08A99 Algebraic structures
08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems
06A12 Semilattices
20N05 Loops, quasigroups
51A25 Algebraization in linear incidence geometry
52A01 Axiomatic and generalized convexity
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