Algebra Colloquium c  2011 AMSS CAS & SUZHOU UNIV Algebra Colloquium 18 : 4 (2011) 611–628 Minimal Characteristic Algebras for Rectangular k-Normal Identities ∗ K. Hambrook S.L. Wismath † Department of Mathematics and Computer Science University of Lethbridge, Lethbridge, Ab., T1K-3M4, Canada E-mail: wismaths@uleth.ca Received 1 February 2008 Revised 13 May 2008 Communicated by K.P. Shum Abstract. A characteristic algebra for a hereditary property of identities of a fixed type τ is an algebra A such that for any variety V of type τ , we have A∈ V if and only if every identity satisfied by V has the property p. This is equivalent to A being a generator for the variety determined by all identities of type τ which have property p. Plonka has produced minimal (smallest cardinality) characteristic algebras for a number of hereditary properties, including regularity, normality, uniformity, biregularity, right- and leftmost, outermost, and external-compatibility. In this paper, we use a construction of P lonka to study minimal characteristic algebras for the property of rectangular k-normality. In particular, we construct minimal characteristic algebras of type (2) for k-normality and rectangularity for 1 ≤ k ≤ 3. 2000 Mathematics Subject Classification: 08A05, 08B05 Keywords: hereditary property of identities, characteristic algebra, k-normal identities, rectangular identities 1 Introduction Let τ be a fixed type of algebras and identities. A property p of identities of type τ is said to be hereditary if any identity deducible (by the usual rules of deduction) from a set of identities all having the property p must also have the property p. A variety V is said to have property p when all the identities of V have property p. When p is a hereditary property, the set p(τ ) of all identities of type τ having property p is an equational theory, and thus defines a variety V p which is the smallest variety to have property p. A characteristic algebra for a hereditary property p is an algebra A such that for any variety V of type τ , every identity of V has property p if and only if A is in V . This is equivalent to saying that the set Id A of identities satisfied by A is * Research supported by NSERC of Canada. † Corresponding author.