Differential Modes

Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until M. Stronkowski and D. Stanovský’s recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes that are not embeddable into semimodules, including structural investigations and an analysis of the lattice of varieties. It is well known that each entropic groupoid (“medial” in the terminology of Ježek-Kepka) with surjective operation embeds as a subreduct into a semimodule over a commutative semiring [1]. In particular, each idempotent and entropic groupoid, i.e. each groupoid mode (as defined e.g. in [12]) embeds into such a semimodule. (See [1] and [6]). Surprisingly, this is no longer true for modes with operations of larger arity. As shown by M. Stronkowski [17] and [18], a mode embeds as a subreduct into a semimodule over a commutative semiring if and only if it satisfies the so-called Szendrei identities. A simpler proof was then given by D. Stanovský [16]. Stronkowski also proved that free modes do not satisfy the Szendrei identities, while Stanovský [16] provided a 3-element example of a mode with one ternary operation (Example 1.1). In this paper we analyze Stanovský’s example, and show that it belongs to the variety of so-called ternary differential modes, Date: September 23, 2007. 2000 Mathematics Subject Classification. 03C05, 08A05, 08B15, 08B20.