DSpace Kolekcja:http://hdl.handle.net/11320/88802020-11-28T23:39:44Z2020-11-28T23:39:44ZFormal Development of Rough Inclusion FunctionsGrabowski, Adamhttp://hdl.handle.net/11320/90192020-04-20T10:37:32Z2019-01-01T00:00:00ZTytuł: Formal Development of Rough Inclusion Functions
Autorzy: Grabowski, Adam
Abstrakt: Rough sets, developed by Pawlak [15], are important tool to describe situation of incomplete or partially unknown information. In this article, continuing the formalization of rough sets [12], we give the formal characterization of three rough inclusion functions (RIFs). We start with the standard one, κ£, connected with Łukasiewicz [14], and extend this research for two additional RIFs: κ 1, and κ 2, following a paper by Gomolińska [4], [3]. We also define q-RIFs and weak q-RIFs [2]. The paper establishes a formal counterpart of [7] and makes a preliminary step towards rough mereology [16], [17] in Mizar [13].2019-01-01T00:00:00ZAIM Loops and the AIM ConjectureBrown, Chad E.Pąk, Karolhttp://hdl.handle.net/11320/90182020-04-20T08:38:33Z2019-01-01T00:00:00ZTytuł: AIM Loops and the AIM Conjecture
Autorzy: Brown, Chad E.; Pąk, Karol
Abstrakt: In this article, we prove, using the Mizar [2] formalism, a number of properties that correspond to the AIM Conjecture. In the first section, we define division operations on loops, inner mappings T, L and R, commutators and associators and basic attributes of interest. We also consider subloops and homomorphisms. Particular subloops are the nucleus and center of a loop and kernels of homomorphisms. Then in Section 2, we define a set Mlt Q of multiplicative mappings of Q and cosets (mostly following Albert 1943 for cosets [1]). Next, in Section 3 we define the notion of a normal subloop and construct quotients by normal subloops. In the last section we define the set InnAut of inner mappings of Q, define the notion of an AIM loop and relate this to the conditions on T, L, and R defined by satisfies TT, etc. We prove in Theorem (67) that the nucleus of an AIM loop is normal and finally in Theorem (68) that the AIM Conjecture follows from knowing every AIM loop satisfies aa1, aa2, aa3, Ka, aK1, aK2 and aK3.
The formalization follows M.K. Kinyon, R. Veroff, P. Vojtechovsky [4] (in [3]) as well as Veroff’s Prover9 files.2019-01-01T00:00:00Z