Journal of Formalized Mathematics
Volume 6, 1994
University of Bialystok
Copyright (c) 1994 Association of Mizar Users

## Preliminaries to Circuits, II

Yatsuka Nakamura
Shinshu University, Nagano
Piotr Rudnicki
University of Alberta, Edmonton
Andrzej Trybulec
Warsaw University, Bialystok
Pauline N. Kawamoto
Shinshu University, Nagano

### Summary.

This article is the second in a series of four articles (started with  and continued in , ) about modelling circuits by many sorted algebras.\par First, we introduce some additional terminology for many sorted signatures. The vertices of such signatures are divided into input vertices and inner vertices. A many sorted signature is called {\em circuit like} if each sort is a result sort of at most one operation. Next, we introduce some notions for many sorted algebras and many sorted free algebras. Free envelope of an algebra is a free algebra generated by the sorts of the algebra. Evaluation of an algebra is defined as a homomorphism from the free envelope of the algebra into the algebra. We define depth of elements of free many sorted algebras.\par A many sorted signature is said to be monotonic if every finitely generated algebra over it is locally finite (finite in each sort). Monotonic signatures are used (see ,) in modelling backbones of circuits without directed cycles.

Partial funding for this work has been provided by: Shinshu Endowment Fund for Information Science, NSERC Grant OGP9207, JSTF award 651-93-S009.

#### MML Identifier: MSAFREE2

The terminology and notation used in this paper have been introduced in the following articles                           

#### Contents (PDF format)

1. Many Sorted Signatures
2. Free Many Sorted Algebras

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