Connections between congruence-lattices and polynomial properties

This paper gives characterisations of properties of varieties such as congruence-distributivity (CD), filtrality (FI), CD and having complemented principal congruences — these last two properties are shown to be equivalent- and having restricted equationally definable principal congruences (REDPC), in terms of the existence of some kind of polynomials. These are generalisations both of Jónsson's famous theorem characterizing CD as well as results concerning the (dual) discriminator. The methods are applied to show that REDPC implies CD, which was a problem asked in [2]. A generalisation of the concept of the Mal'cev condition — the so called Pixley-condition — is defined, and it is shown that filtrality and REDPC are Pixley-conditions. The relations between several concepts connected with the above ones are also investigated.The definitions can be found in section 2, and the results are contained in section 4, section 5, and section 6. We call attention to the figure at the end of the paper, which contains most of our results and gives a survey of the concepts examined.Presented by G. Grätzer.Thanks are also due to Peter Krauss, who called our attention to some gaps in our proofs in a previous version of the present paper.