Pre-image of functions in $C(L)$ | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 3، دوره 15، شماره 1، مهر 2021، صفحه 35-58 اصل مقاله (764.46 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.52547/cgasa.15.1.35 | ||
| نویسندگان | ||
| Ali Rezaei Aliabad1؛ Morad Mahmoudi2 | ||
| 1Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran | ||
| 2Department of of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran | ||
| چکیده | ||
| Let $C(L)$ be the ring of all continuous real functions on a frame $L$ and $S\subseteq{\mathbb R}$. An $\alpha\in C(L)$ is said to be an overlap of $S$, denoted by $\alpha\blacktriangleleft S$, whenever $u\cap S\subseteq v\cap S$ implies $\alpha(u)\leq\alpha(v)$ for every open sets $u$ and $v$ in $\mathbb{R}$. This concept was first introduced by A. Karimi-Feizabadi, A.A. Estaji, M. Robat-Sarpoushi in {\it Pointfree version of image of real-valued continuous functions} (2018). Although this concept is a suitable model for their purpose, it ultimately does not provide a clear definition of the range of continuous functions in the context of pointfree topology. In this paper, we will introduce a concept which is called pre-image, denoted by ${\rm pim}$, as a pointfree version of the image of real-valued continuous functions on a topological space $X$. We investigate this concept and in addition to showing ${\rm pim}(\alpha)=\bigcap\{S\subseteq{\mathbb R}:~\alpha\blacktriangleleft S\}$, we will see that this concept is a good surrogate for the image of continuous real functions. For instance, we prove, under some achievable conditions, we have ${\rm pim}(\alpha\vee\beta)\subseteq {\rm pim}(\alpha)\cup {\rm pim}(\beta)$, ${\rm pim}(\alpha\wedge\beta)\subseteq {\rm pim}(\alpha)\cap {\rm pim}(\beta)$, ${\rm pim}(\alpha\beta)\subseteq {\rm pim}(\alpha){\rm pim}(\beta)$ and ${\rm pim}(\alpha+\beta)\subseteq {\rm pim}(\alpha)+{\rm pim}(\beta)$. | ||
| کلیدواژهها | ||
| Frame؛ pointfree topology؛ $C(L)$؛ pre-image؛ prime ideal and maximal ideal in frames؛ $f$-algebra | ||
| مراجع | ||
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[1] Ball, R.N. and Hager, A.W., On the localic Yoshida representation of an archimedean lattice ordered group with weak order unit, J. Pure Appl. Algebra 70 (1991), 17-43. [2] Banaschewski, B., "The Real Numbers in Pointfree Topology", Textos de Mathematica (Series B), Vol. 12, University of Coimbra, 1997. [3] Banaschewski, B., On the function rings of pointfree topology, Kyungpook Math. J. 48 (2008), 195-206. [4] Blyth, T., "Lattices and Ordered Algebraic Structures", Springer, 2005. [5] Davey, B. and Priestley, H., "Introduction to Lattices and Order", Cambridge University Press, 2002. [6] Ebrahimi, M.M. and Karimi Feizabadi, A., Prime representation of real Riesz maps, Algebra Universalis, 54(3) (2005), 291-299. [7] Estaji, A.A., Karimi Feizabadi, A. and Emamverdi, B., Representation of real Riesz maps on a strong f-ring by prime elements of a frame, Algebra Universalis 79(14) (2018). [8] Estaji, A.A., Robat Sarpoushi, M., and Elyasi, M., Further thoughts on the ring Rc(L) in frames, Algebra Universalis, 80(43) (2019). [9] Estaji, A.A., Karimi Feizabadi, A., and Abedi, A., Zero sets in pointfree topology and strongly z-ideals, Bull. Iranian Math. Soc. 41(5) (2015), 1071-1084. [10] Gillman, L. and Jerison, M., "Rings of Continuous Functions", Springer-Verlag, 1976. [11] Gratzer, G., "General Lattice Theory", Birkhauser, 1998. [12] Johnstone, P.T., "Stone Spaces", Cambridge Univ. Press, 1982. [13] Karimi Feizabadi, A., Estaji, A.A., and Robat-Sarpoushi, M., Pointfree version of image of real-valued continuous functions, Categ. Gen. Algebr. Struct. Appl., 9(1) (2018), 59-75. [14] Picado, J. and Pultr, A., "Frames and Locales: topology without points", Frontiers in Mathematics, Springer, 2012. [15] Roman, S., "Lattices and Ordered Sets", springer, 2008. | ||
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