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Let [Formula: see text] be a finite alphabet and [Formula: see text] the set of all words over [Formula: see text]. A subword-free language (also known as a hypercode) is an independent subset of [Formula: see text] with respect to the embedding order (denoted by [Formula: see text]) on [Formula: see text]. In this paper we introduce three subsets of the partial order [Formula: see text], and study three subclasses of languages defined by these subsets. They are out subword-free languages, left subword-free languages and right subword-free languages. The properties of these languages are established for determining their combinatorial and algebraic structures. By equipping them with two binary operations, respectively, all these classes of languages form semilattice-ordered semigroups. It is shown that they are freely generated by [Formula: see text] in three subvarieties of semilattice-ordered semigroups, respectively. It is also shown that the word problems for these free algebras are solvable.

No abstract

A semiring variety is d-semisimple if it is generated by the distributive lattice of order two and a finite number of finite fields. A d-semisimple variety V = HSP{B2, F1,..., Fk} plays the main role in this paper. It will be proved that it is finitely based, and that, up to isomorphism, the two-element distributive lattice B2 and all subfields of F1,..., Fk are the only subdirectly irreducible members in it. [The first author is supported by China Postdoctoral Science Foundation, Grant 2011M501466 and the Natural Science Foundation of Shannxi Province, Grant 2011JQ1017. The second authoris Supported by the Ministry of Education, Science and Technological Development of Serbia,Grant 174018. The third author is Supported by the Ministry of Education, Science and TechnologicalDevelopment of Serbia, Grant 174026]

Up to isomorphism, there are 61 ai-semirings of order three. The finite basis problem for these semirings is investigated. This problem for 45 semirings of them is answered by some results in the literature. The remaining semirings are studied using equational logic. It is shown that with the possible exception of the semiring \(S_7\), all ai-semirings of order three are finitely based.