Abstract
LetL(K) denote the lattice (ordered by inclusion) of quasivarieties contained in a quasivarietyK and letD 2 denote the variety of distributive (0, 1)-lattices with 2 additional nullary operations. In the present paperL(D 2) is described. As a consequence, ifM+N stands for the lattice join of the quasivarietiesM andN, then minimal quasivarietiesV 0,V 1, andV 2 are given each of which is generated by a 2-element algebra and such that the latticeL(V 0+V1), though infinite, still admits an easy and nice description (see Figure 2) while the latticeL(V 0+V1+V2), because of its intricate inner structure, does not. In particular, it is shown thatL(V 0+V1+V2) contains as a sublattice the ideal lattice of a free lattice with ω free generators. Each of the quasivarietiesV 0,V 1, andV 2 is generated by a 2-element algebra inD 2.
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Adams, M.E., Dziobiak, W. Joins of minimal quasivarieties. Stud Logica 54, 371–389 (1995). https://doi.org/10.1007/BF01053005
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DOI: https://doi.org/10.1007/BF01053005
Key words
- Quasivariety
- lattice of quasivarieties
- distributive (0, 1)-lattice
- nullary operations
- free lattice
- Priestley space
- graph