Embedding Modes into Semimodules, Part I

By recent results of M. Stronkowski, it is known that not all modes embed as subreducts into semimodules over commutative unital semirings. Related to this problem is the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provide a general construction of such semirings, along with basic examples and some general properties. The second part of the paper will deal with applications of the general construction to some selected varieties of modes, and will provide a description of semirings determining varieties of semimodules having algebras from these varieties as idempotent subreducts. Modes are idempotent and entropic algebras. (See [10, 13] for a general introduction and basic theory.) One of the main methods of representing modes is given by embedding them as subreducts into modules over commutative rings, and more generally into semimodules over commutative semirings. Since modes are idempotent algebras, the embedding is in fact into the full idempotent reducts of such (semi)modules. In the case of modules, such full idempotent reducts are actually affine spaces (sometimes known as affine modules). For a given ring they form a variety of Mal’cev modes. In the case of semimodules over a given semiring, such full idempotent reducts, known as semi-affine spaces (or sometimes affine semimodules), do not form a quasivariety in general, and may be trivial. A general characterization of modes in a given variety that embed as subreducts (subalgebras of reducts) into an affine space is given in [13, Theorem 7.2.3]. There are some other results showing that modes satisfying certain additional conditions are subreducts of affine spaces, see [13, Chapter 7], [11, 12, 10]. However, there is no easy general method known of deciding for an arbitrary mode if it is embeddable into an affine space. On the other Date: October 4, 2010. 2000 Mathematics Subject Classification. 08A62, 08A05, 08B99, 08C05, 16Y60.