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Embedding lattices in lattices of varieties of groups
Izvestiya: Mathematics, 1999Let \(\Lambda\) denote the direct product of subspace lattices, one for each finite-dimensional vector space over the \(2\)-element field, and let \(\mathbb{A}_2\) be the group variety defined by the law \(x^2=1\). The main result of the paper is that \(\Lambda\) embeds in the interval \([\mathbb{A}_2^4,\mathbb{A}_2^5]\) of the lattice of group ...
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Varieties of lattice-ordered algebras
Algebra Universalis, 1983The variety of lattice-ordered groups generated by fully ordered groups is axiomatised either by (i) \((x\vee y)^ 2=x^ 2\vee y^ 2\) or (ii) \((x\vee 1)\wedge y^{-1}(x^{-1}\vee 1)y=1\). The variety of lattice- ordered loops generated by fully ordered loops is axiomatised by: (a) \((x/z\vee 1)\wedge y\backslash((z/x\vee 1)y)=1\) (b) \((x/z\vee 1)\wedge(((
Evans, T., Hartmann, P. A.
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1994
As it was mentioned in Section 1 of Chapter 9, the set of all 1-varieties L, is a complete lattice where, for any l-varieties X, Y, the greatest lower bound, or meet, of X, Y is their set-theoretic intersection and their least upper bound, or join, is the intersection of all l-varieties containig both X and Y; more formally $$x\mathop \wedge ...
V. M. Kopytov, N. Ya. Medvedev
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As it was mentioned in Section 1 of Chapter 9, the set of all 1-varieties L, is a complete lattice where, for any l-varieties X, Y, the greatest lower bound, or meet, of X, Y is their set-theoretic intersection and their least upper bound, or join, is the intersection of all l-varieties containig both X and Y; more formally $$x\mathop \wedge ...
V. M. Kopytov, N. Ya. Medvedev
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Semicomplements in lattices of varieties
Algebra Universalis, 1992A lattice \(L\) with 0 and 1 is said to be upper semicomplemented if, for each \(x\in L\backslash\{0\}\), there exists \(y\in L\backslash\{1\}\) with \(x\vee y=1\). The author and \textit{M. V. Volkov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1982, No.
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Varieties of Lattice-Ordered Groups
1989We begin by recalling a fundamental theorem due to Birkhoff as shown in [Burris and Sankappanavar, 1981; Theorem II.11.9].
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A variety of lattices whose quasivarieties are varieties
Algebra Universalis, 1978Grätzer, G., Lakser, H.
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