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In this paper, first the congruences in the category PosAct-S of all poset acts over a pomonoid S; an S-act in the category Pos of all posets, with action preserving monotone maps between them, are introduced. Then, we study the existence of the free and cofree objects in the category PosAct-S. More precisely, we consider all forgetful functors between this category and the categories Pos-S of all S-posets, Pos of all posets, Act-S of all S-acts, and Set of all sets, and we study the existence of their left and right adjoints. It is shown that the category Pos-S is a full reflective and coreflective subcategory of PosAct-S.
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Bloom, S.L., Varieties of ordered algebras, J. Comput. System Sci. 13 (1976), 200-212.
S. Bulman-Fleming, Subpullback flat S-posets need not be subequlizer flat, Semigroup Forum 78(1) (2009), 27-33.
S. Bulman-Fleming and V. Laan, Lazard’s theorem for S-posets, Math. Nachr. 278(15) (2005), 1743-1755.
S. Bulman-Fleming, A. Gilmour and M. Kilp, Flatness properties of S-posets, Comm. Algebra 34 (2006), no. 4, 1291-1317.
S. Bulman-Fleming and M. Mahmoudi, The category of S-posets, Semigroup Forum 71(3) (2005), 443-461.
Blyth, T. S. and Janowitz, M. F., Residuation Theory, Pergamon Press, Oxford, 1972.
H. Ehrig, F. Parisi-Presicce, P. Boehm, C. Rieckhoff, C. Dimitrovici and M. Grobe-Rhode, Algebraic data type and process specifications based on projection spaces, Lecture Notes in Computer Science, 332, Springer, Berlin-Heidelberg, 1988.
Fakhruddin, S.M., Absolute retractness and amalgams in pomonoids, Semigroup Forum, 33 (1986), 15-22.
Fakhruddin, S. M., On the category of S-posets, Acta Sci. Math. (Szeged) 52 (1988), 85-92.
Kilp, M., U. Knauer, A. Mikhalev, Monoids, Acts and Categories, Walter de Gruyter, Berlin, New York, 2000.
Rasouli, H., Categorical properties of regular monomorphisms of S-posets, European J. of pure and Applied Math. 7(2) (2014), 166-178.
Rasouli, H., Barzegar, H., The regular injective envelope of S-posets, Semigroup Forum, 92(1) (2015), 186-197.
Shahbaz, L., Mahmoudi, M., Various kinds of regular injectivity for S-posets, Bull. Iranian Math. Soc. 40(1) (2014), 243-261.
Shahbaz, L., Mahmoudi, M., Injectivity of S-posets with respect to down closed regular monomorphisms, Semigroup Forum 91 (2015), 584-600.
Skornyakov, L.A., A general view of representations of monoids, (Russian) Algebra i Logika 20 (1981), 571-574, 600.
Skornyakov, L.A., Injective objects of categories of representations of monoids, (Russian) Mat. Zametki 36 (1984), 159-170.
Skornjakov, L.A., On the injectivity of ordered left acts over monoids, Vestnik Moskov. univ. Ser. I Math. Mekh. (1986) 17-19 (in Russian).