An Abstract Algebraic Theory of L-Fuzzy Relations for Relational Databases

Classical relational databases lack proper ways to manage certain real-world situations including imprecise or uncertain data. Fuzzy databases overcome this limitation by allowing each entry in the table to be a fuzzy set where each element of the corresponding domain is assigned a membership degree from the real interval [0…1]. But this fuzzy mechanism becomes inappropriate in modelling scenarios where data might be incomparable. Therefore, we become interested in further generalization of fuzzy database into L-fuzzy database. In such a database, the characteristic function for a fuzzy set maps to an arbitrary complete Brouwerian lattice L. From the query language perspectives, the language of fuzzy database, FSQL extends the regular Structured Query Language (SQL) by adding fuzzy specific constructions. In addition to that, L-fuzzy query language LFSQL introduces appropriate linguistic operations to define and manipulate inexact data in an L-fuzzy database. This research mainly focuses on defining the semantics of LFSQL. However, it requires an abstract algebraic theory which can be used to prove all the properties of, and operations on, L-fuzzy relations. In our study, we show that the theory of arrow categories forms a suitable framework for that. Therefore, we define the semantics of LFSQL in the abstract notion of an arrow category. In addition, we implement the operations of L-fuzzy relations in Haskell and develop a parser that translates algebraic expressions into our implementation.

Abstract of title of part of Lots no. 12 and 13 in the 8th Concession in the Township of Grantham

Abstract of title of parts of lots nos. 12 and 13 in the 8th Concession in the Township of Grantham – Orsen Phelps to Owen Cifford and abstract of title of parts of Lots nos. 10 and 11 in the 9th Concession in the Township of Grantham – David Grass to Owen Clifford. This is a handwritten 2 page summary. This is part of the Owen Clifford Estate signed by W.W. Powell, registrar, Feb. 20, 1871.

Application for Loan, Mortgage Loan Envelope, and Abstract of Title

Atkins, Sarah A. and Carlisle A. Gardner includes: Application for Loan, July 26, 1884; Mortgage Loan Envelope no. 255 for July 1, 1884 – July 1, 1889 and Abstract of Title, August 28, 1884.

Crew, Isaac M., includes: Application for Loan, Mortgage Loan Envelope, and Abstract of Title

Crew, Isaac M., includes: Application for Loan, March 10, 1885; Mortgage Loan Envelope no. 679 for Jan. 1, 1885 – Jan. 1, 1890 and Abstract of Title, April 13, 1885.

Dennis, George W. and Mary E. Dennis, includes: Application for Loan on Real Estate, and Abstract of Title

Dennis, George W. and Mary E. Dennis, includes: Application for Loan on Real Estate, July 10, 1882; and Abstract of Title, Sept. 2, 1882.

Schmidt, John, includes: Application for Loan, and Mortgage Loan Envelope, and Abstract of Title

Schmidt, John, includes: Application for Loan, Aug. 8, 1884 and Mortgage Loan Envelope no. 302 for July 1, 1884 – July 1, 1889 and Abstract of Title, Sept. 6, 1884.

Letter which was enclosed with the application, bond mortgage and abstract of the Jones and Mater accounts to Samuel Woodruff from Jarvis, Conklin and Co.

Letter which was enclosed with the application, bond mortgage and abstract of the Jones and Mater accounts to Samuel Woodruff from Jarvis, Conklin and Co., May 4, 1882.

Voucher #7 from the Engineer Department of Port Dalhousie and Thorold Railway Extension to W.G. Thompson accompanied by an abstract of vouchers and summaries of account paid by W.G. Thompson for surveys, Jan. 31, 1857

Voucher #7 from the Engineer Department of Port Dalhousie and Thorold Railway Extension to W.G. Thompson accompanied by an abstract of vouchers and summaries of account paid by W.G. Thompson for surveys, Jan. 31, 1857.

Voucher #9 from the Engineer Department of Port Dalhousie and Thorold Railway Extension to W. G. Thompson accompanied by an abstract of vouchers for surveys, Feb. 28, 1857

Voucher #9 from the Engineer Department of Port Dalhousie and Thorold Railway Extension to W. G. Thompson accompanied by an abstract of vouchers for surveys, Feb. 28, 1857

Abstract of the forging

Abstract of the forging (1 page, handwritten) n.d.

Abstract of estimates

Abstract of estimates, June 13, 1856.

Abstract of estimates given to Andrew Mains for ditching on the main drain of marsh lands for 1855 and 1856

Abstract of estimates given to Andrew Mains for ditching on the main drain of marsh lands for 1855 and 1856. This is signed by S.D. Woodruff, May 1, 1857.

Abstract of tenders received for the Welland Canal

Abstract of tenders received for the Welland Canal, Sept. 23, 1847.

Abstract of quantity of excavation for lock no.12

Abstract of quantity of excavation for lock no.12, July 4, 1854.

List (1 page, handwritten) of Sarnia Branch for which no abstract of the title has been furnished

List (1 page, handwritten) of Sarnia Branch for which no abstract of the title has been furnished. This was signed by H. [illegible] in 1839 and a note was added to this list on Nov. 15, 1859.