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The algebra of mode homomorphisms. (English) Zbl 1312.08001
A mode is an algebra in which every singleton is a subalgebra and every operation is a homomorphism (all operations commute). Both properties can be formulated as satisfying some identities in an algebra. In this paper, algebras of mode homomorphisms are introduced and studied in detail. They are constructed as follows: for modes in a prevariety \(K\) of modes of given (plural) type \(\tau\colon\Omega\to\mathbb Z_+\), one considers the set of all homomorphisms from subalgebras of one mode to subalgebras of another with operations from \(\Omega\) defined in a special but natural way to obtain again an algebra in \(K\).

MSC:
08A05 Structure theory of algebraic structures
06A12 Semilattices
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[1] Bošnjak I., Madarász R., Retraction closure property, Algebra Universalis, 2013, 69(3), 279-285 http://dx.doi.org/10.1007/s00012-013-0229-0
[2] Csákány B., Varieties of affine modules, Acta Sci. Math. (Szeged), 1975, 37, 3-10
[3] Hion Ja.V., Ω-ringoids, Ω-rings and their representations, Trudy Moskov. Mat. Obšč., 1965, 14, 3-47 (in Russian)
[4] Mašulović D., Turning retractions of an algebra into an algebra, Novi Sad J. Math., 2004, 34(1), 89-98 · Zbl 1107.08003
[5] Neumann W.D., On the quasivariety of convex subsets of affine spaces, Arch. Math. (Basel), 1970, 21, 11-16 http://dx.doi.org/10.1007/BF01220869 · Zbl 0194.01502
[6] Pilitowska A., Zamojska-Dzienio A., Varieties generated by modes of submodes, Algebra Universalis, 2012, 68(3-4), 221-236 http://dx.doi.org/10.1007/s00012-012-0201-4 · Zbl 1270.08004
[7] Pöschel R., Reichel M., Projection algebras and rectangular algebras, In: General Algebra and Applications, Potsdam, January 31-February 2, 1992, Res. Exp. Math., 20, Heldermann, Berlin, 1993, 180-194
[8] Romanowska A.B., Smith J.D.H., Modal Theory, Res. Exp. Math., 9, Heldermann, Berlin, 1985
[9] Romanowska A.B., Smith J.D.H., Modes, World Scientific, Singapore, 2002 http://dx.doi.org/10.1142/4953
[10] Smith J.D.H., Romanowska A.B., Post-Modern Algebra, Pure Appl. Math. (N.Y.), John Wiley & Sons, New York, 1999 http://dx.doi.org/10.1002/9781118032589 · Zbl 0946.00001
[11] Sokratova O., Kaljulaid U., Ω-rings and their flat representations, In: Contributions to General Algebra, 12, Vienna, June 3-6, 1999, Heyn, Klagenfurt, 2000, 377-390 · Zbl 0971.16024
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