Enriched Płonka sums

The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider the question of when this quasivariety is a variety. The main result asserts that if $$\mathcal {V}$$ is a strongly irregular variety with no nullary operations and at least one non-unary operation, and $$\mathcal {S}$$ is the variety, of the same type as $$\mathcal {V}$$ , equivalent to the variety of semilattices, then the Mal’tsev product $$\mathcal {V}\circ \mathcal {S}$$ is a variety. It consists precisely of semilattice sums of algebras in $$\mathcal {V}$$ . We derive an equational base for the product from an equational base for $$\mathcal {V}$$ . However, if $$\mathcal {V}$$ is a regular variety, then the Mal’tsev product may not be a variety. We discuss various applications of the main result, and examine some detailed representations of algebras in $$\mathcal {V}\circ \mathcal {S}$$ .

Previous article Next article Conditions for Positive and Nonnegative Definiteness in Terms of PseudoinversesArthur AlbertArthur Alberthttps://doi.org/10.1137/0117041PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Richard Bellman, Introduction to matrix analysis, McGraw-Hill Book Co., Inc., New York, 1960xx+328 MR0122820 0124.01001 Google Scholar[2] A. Ben-Israel and , A. Charnes, Contributions to the theory of generalized inverses, J. Soc. Indust. Appl. Math., 11 (1963), 667–699 10.1137/0111051 MR0179192 0116.32202 LinkISIGoogle Scholar[3] T. N. E. Greville, Some applications of the pseudoinverse of a matrix, SIAM Rev., 2 (1960), 15–22 10.1137/1002004 MR0110185 0168.13303 LinkISIGoogle Scholar[4] Charles A. Rohde, Generalized inverses of partitioned matrices, J. Soc. Indust. Appl. Math., 13 (1965), 1033–1035 10.1137/0113070 MR0190161 0145.03801 LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Error Analysis of Multirate Leapfrog-Type Methods for Second-Order Semilinear OdesConstantin Carle and Marlis HochbruckSIAM Journal on Numerical Analysis, Vol. 60, No. 5 | 13 October 2022AbstractPDF (1129 KB)The truncated moment problem on the union of parallel linesLinear Algebra and its Applications, Vol. 649 | 1 Sep 2022 Cross Ref On the non-symmetric semidefinite Procrustes problemLinear Algebra and its Applications, Vol. 648 | 1 Sep 2022 Cross Ref Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation riskJournal of Industrial & Management Optimization, Vol. 18, No. 1 | 1 Jan 2022 Cross Ref Stability and Functional Superconvergence of Narrow-Stencil Second-Derivative Generalized Summation-By-Parts DiscretizationsJournal of Scientific Computing, Vol. 90, No. 1 | 10 December 2021 Cross 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