A well known theorem of Murskiı̆’s asserts that almost every finite, nonunary algebra is idemprimal. We derive an analagous result under the assumption that all basic operations are idempotent. If the algebra contains a basic l-ary idempotent operation with l > 2 then the algebra is idemprimal with probability 1. However, for an algebra with a single basic binary operation, the probability of id...

A subgroup H of a finite group G is said to be W -S-permutable in G if there is a subgroup K of G such that G = HK and H ∩K is a nearly S-permutable subgroup of G. In this article, we analyse the structure of a finite group G by using the properties of W -S-permutable subgroups and obtain some new characterizations of finite p-nilpotent groups and finite supersolvable groups. Some known results...

One of the important classes of varieties identified in tame congruence theory is the class of varieties which are n-permutable for some n. In this paper we prove two results: (1) For every n > 1 there is a polynomial-time algorithm which, given a finite idempotent algebra A in a finite language, determines whether the variety generated by A is n-permutable; (2) A variety is n-permutable for so...

A diagrammatic statement is developed for the generalized semidistributive law in case of single algebras assuming that their congruences are permutable. Without permutable congruences, a diagrammatic statement is developed for the ∧-semidistributive law.

We present a new and useful congruence identity satisfied by m-permutable varieties. It has been proved in [L1] that every m-permutable variety satisfies a non-trivial lattice identity (depending only on m). In [L2] we have found another interesting identity: Theorem 1. For m ≥ 3, every m-permutable variety satisfies the congruence identity αβh = αγh, for h = m[ m+1 2 ]− 1 Here, [ ] denotes int...

Brevity is the soul of wit. William Shakespeare A cyclically permutable code is a set of codewords having the property that no codeword is a cyclic shift of another codeword. We study the problem of constructing cyclically permutable codes of large size and low correlation. Cyclically permutable codes are used in code-division multiple-access systems realized by e.g. direct-sequence modulation ...

A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. An important class of these codes are the constant weight cyclically permutable codes. In a code of this class all codewords have the same weight w. These codes have many applications, including in optical code-division multiple-access communication systems and in constructing proto...

Y. Li gave a characterization of the class of finite soluble groups in which every subnormal subgroup is normal by means of NE -subgroups: a subgroup H of a group G is called an NE -subgroup of G if NG(H) ∩ H = H. We obtain a new characterization of these groups related to the local Wielandt subgroup. We also give characterizations of the classes of finite soluble groups in which every subnorma...