By Sándor Dominich

This ebook takes a different method of details retrieval through laying down the rules for a contemporary algebra of data retrieval in keeping with lattice conception. All significant retrieval equipment built thus far are defined intimately – Boolean, Vector area and probabilistic tools, but in addition internet retrieval algorithms like PageRank, HITS, and SALSA – and the writer exhibits that all of them could be taken care of elegantly in a unified formal manner, utilizing lattice concept because the one uncomplicated thought. extra, he additionally demonstrates that the lattice-based method of info retrieval permits us to formulate new retrieval methods.

Sándor Dominich’s presentation is characterised by way of an engineering-like process, describing all equipment and applied sciences with as a lot arithmetic as wanted for readability and exactness. His readers in either computing device technology and arithmetic will learn the way one unmarried idea can be utilized to appreciate crucial retrieval tools, to suggest new ones, and in addition to realize new insights into retrieval modeling typically. therefore, his ebook is acceptable for researchers and graduate scholars, who will also enjoy the many workouts on the finish of every chapter.

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**Example text**

6 shows a 6-element poset that is not a lattice because there are multiple infima (maxima). Fig. 6. A poset that is not a lattice. 5 Complete, Atomic Lattice A lattice L is called complete if every nonempty subset of L has a supremum and an infimum. Any complete lattice L has a smallest element, denoted by 0, and a largest element, denoted by 1. It can be immediately seen that every finite lattice is complete. 2 • The lattice (L, ≤) on the set L = {0, x, y, z, 1} defined by the ordering relation {(0, x), (0, z), (x, y), (y, 1), (z, 1)} is represented by the Hasse diagram in Fig.

Since L is modular, using Eq. 13), we find that it follows that S is modular as well. 8 Distributive Lattice We have seen that the weak distributivity property [Eq. 5)] as well as its dual hold in any lattice. We may have a lattice with the relation < (instead of ≤). , for equality), then nothing can be said about its dual (because modularity is not a consequence of the axioms defining a lattice). 14) A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C). Let us show the part of this equivalence. We have (A ∧ B) ∨ (A ∧ C) = ((A ∧ B) ∨ A) ∧ ((A ∧ B) ∨ C) = A ∧ ((A ∧ B) ∨ C) = [left part of Eq.

Compatibility of relevance assessments. Vector space method as lattice-lattice mapping. , they look like (or are equivalent to) the following drawing: 24 1 Introduction Further, we show that the very character of the underlying mechanism of vector space retrieval is nonsubmodularity. , way of reasoning) is characterized by orthomodularity, the application of lattices in vector space retrieval is characterized by nondistributivity and nonsubmodularity. , their entities are not always compatible with one another).