Algorithms used by the UACalcator
Finding Cg(a,b): An (essentially) linear time algorithm for this
is given in [Freese 2008],
based on the work in the early 1970's of R. Tarjan and others.
Finding the TCT type set of an algebra: We follow a slightly modified version of the algorithm described in [Berman, Kiss, Prohle, Szendrei]. Also see my talk on this subject, which shows that algorithm runs in \(\\mathbf A\^3\) time.
Bounding the TCT type set of \(\mathbf V(A)\): This is explained here.
The Maltsev condition for a locally finite variety to have a difference term, Ross Willard. This is explained here.
Testing primality: We follow the procedure given in [Clark, Davey, Pitkethly, Rifqui]. Let 0 and 1 be two elements of A. They show that A is primal if
 the \(n\) characteristic functions (1 at \(i\), 0 elsewhere) are \(\mathbf F(1)\)
 the subalgebra of \(\mathbf F(1)\) generated by these characteristic functions includes the identity map.
 the subalgebra of \(\mathbf A^4\) generated by (0,1,0,1) and (0,0,1,1) contains (0,0,0,1) [the meet operation].
References
 J. Berman, E. Kiss, P. Prohle, and A. Szendrei, The type set of a finitely generated variety, Discrete Math. 112(1993), 120.
 D. Clark, B. Davey, J. Pitkethly, D. Rifqui, Flat unars: the primal, the semiprimal and the dualisable, Algebra Universalis, 63(2010), 303329.
 Ralph Freese, Computing Congruences Efficiently, Algebra Universalis, 59(2008), 337343.
 R. Freese, Automated Lattice Drawing, Lecture Notes in Artificial Intelligence, 2961, Springer, Berlin, 2004, 112127.
 R. Freese, Notes on the Birkhoff Construction, pdf .
 R. Freese, Bounds on the TCT type set of \(\mathbf V(A)\), \(A\) idempotent, pdf .
 R. Freese, Notes on centrality, pdf .

Ralph Freese and Ralph McKenzie,
Commutator
theory for congruence modular varieties, 2^{nd} edition.
A revised, online version.
 Ralph Freese and Matthew Valeriote, On the complexity of some Maltsev conditions, Intern. J. Algebra & Computation, 19(2009), 4177.
 D. Hobby and R. McKenzie,
The Structure of Finite Algebras
Comtemporary Math.,
AMS, Providence, RI, 1988.
This is online:
http://www.ams.org/online_bks/conm76/. There is also the
complete book here.