## 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