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This book addresses the notoriously difficult problem of characterizing lattices of subquasivarieties of a quasivariety.Not every lattice can be represented as a lattice of subquasivarieties. Of those that can be represented, some cannot be represented in languages with equality.

Exploration of the connection between lattices of algebraically closed subsets and subquasivariety lattices yields new avenues for determining if a lattice can be represented as a lattice of subquasivarieties.Both longstyle representations and the new strong shortstyle representations guarantee representations in a language with equality.The two complementary methods not only generate extensive new families of examples, but also reveal structural constraints previously unknown.

Highlights of the book include a detailed summary of the current knowledge on properties of subquasivariety lattices with emphasis on equaclosure operators; detailed examples of longstyle and strong shortstyle representations and methods for creating lattices that can be represented in a language with equality; necessary conditions on an equaclosure operator on a lattice for a representation to exist; techniques for creating new longstyle representable lattices from existing representations; a thorough discussion on the representability of distributive lattices; a new equaclosure operator property required for a represenation; and a new lattice theoretic property required for a representation.A concluding collection of open problems highlights the many avenues still left to explore, inviting readers to advance the field further.

This book is ideal for graduate students in mathematics with interests in logic, universal algebra, or lattice theory. Researchers will find the new techniques and results indispensable tools for probing the structure of subquasivarieties lattices.