The Structure of a Graph Inverse Semigroup

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extended to a homomorphism (that preserves zero) of the corresponding graph inverse semigroups if and only if it is injective.Comment: 19 pages; corrected errors, improved organization, strengthened a result (Theorem 20), added reference

Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)

We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1.Comment: 5 pages. To appear, Journal de Theorie des Nombres de Bordeau

Belgian corporate finance in a European perspective

When analysing corporate finance, it is necessary to take account of various factors which may cause significant disparities between firms, such as their size and their sector of activity. Taking account of the size aspect, by neutralising sectoral disparities, there are few differences between the debt levels of small, medium-sized and large firms. Conversely, the debt structure appears to depend on the firm’s size : small firms are more dependent on bank loans. That is confirmed by the high degree to which they make use of credit facilities. Nonetheless, surveys indicate that access to finance is not a major constraint for SMEs, be they Belgian or European : they perceive access to finance, and more specifically access to bank finance, as relatively easy. In contrast, the financial structure of firms differs widely between sectors, and depends to a great extent on the associated intrinsic activity and the scale of the investments. Sectors with high investment ratios, such as the transport and communication sector or the energy sector, mainly use long-term finance. Ample equity capital enables them to maintain a balanced financial situation. Conversely, highly labour-intensive sectors, such as construction or trade, display much higher debt-to-equity ratios ; their debts are mainly short term and they make extensive use of trade credit. A more detailed analysis of the manufacturing sector also reveals differences of financial structure between firms which are classed as innovative and those which are not. In particular, if the chemical industry is excluded, the firms in the innovative sectors make less use of bank loans and record more short-term debt than firms in non-innovative sectors. That may reflect the lenders’ desire to limit the risk incurred, particularly by using the threat of non-renewal of the loan to encourage the manager to behave efficiently. The qualitative surveys appear to indicate that the financial constraint is felt more by innovative firms than by SMEs in general. That expresses a financing need specific to innovative SMEs. At the early stages in their development, they depend almost exclusively on the entrepreneur’s personal resources and those of his friends and family, and venture capital only takes over in the later stages. Finally, as regards the financing structure, a comparison between Belgian firms and their European counterparts, after neutralising the specific effects of size and sector, indicates that the former issue larger amounts of capital. Abundant intra-group financial flows and a favourable institutional context are conducive to that situation.bank lending, venture capital, corporate finance.

The financial situation of non-financial corporations

As entities where the production process takes place, non financial corporations deserve full attention in the analysis of the real and financial accounts. However, the examination of the financial behaviour of non financial corporations is traditionally confined to their financing : thus, the Bank’s annual report generally only considers their liabilities. Yet the formation of financial assets by non financial corporations is a significant item of information, in both statistical and economic terms. The total financial assets held by non financial corporations in Belgium are considerable, as they far exceed the total financial assets of households. A study recently revealed the historically high level of financial asset formation by non financial corporations in the main industrialised countries, a factor which could help to explain the relatively low level of long-term interest rates. In recent years, Belgian non financial corporations have formed substantially more financial assets than their counterparts in the euro area. In 2005, the financial assets held by Belgian companies were – partly for that reason – almost double the financial assets, expressed in percentages of GDP, of non financial corporations in the euro area. However, this situation certainly does not indicate any risk aversion on the part of Belgian non financial corporations. In fact the liabilities side of their balance sheets also records growth of new financial liabilities in excess of the figure for non financial corporations in the euro area, so that – in terms of liabilities, too – the outstanding total in Belgium is comparatively much higher than the outstanding total in the euro area. This paradoxical situation of Belgian non financial corporations – the fact that they hold much more substantial assets while at the same time contracting significantly greater liabilities – is due to the fact that the Belgian non financial corporations sector includes coordination centres and non financial holding companies, i.e. undertakings acting, as it were, as financial intermediaries. If the coordination centres and non financial holding companies are excluded, the overall outstanding total gross assets and liabilities of Belgian non financial corporations reverts to a level very close to the figure for the euro area. Finally, in 2005 Belgian non financial institutions recorded an outstanding total of net liabilities amounting to around 100 p.c. of GDP, i.e. slightly more than the outstanding total for non financial corporations in the euro area. It is also apparent that the exclusion of the coordination centres and non financial holding companies has hardly any effect on Belgium’s net debtor position, confirming that these two types of institutions perform an intermediary function. Although it is not possible to isolate the coordination centres and non financial holding companies in the case of new transactions (flows), there is no reason to think that the effect would be any different : there would be a significant decline in the gross flows of new assets and liabilities, but that would have a negligible influence on the financial balance. That balance is very similar to the one for the euro area.flow of funds, corporate finance

The Belgian deposit guarantee scheme in a European perspective

During the recent financial crisis, the deposit guarantee scheme in Belgium – as in other European countries – played a role in preventing bank runs and restoring confidence : to that end, the intervention ceilings were raised substantially and the scope of the scheme was extended to include certain life insurance policies. Finally, the expansion of the system’s coverage had to be financed by a sharp increase in the contributions from financial institutions. First of all, that measure had a positive impact on the budget ; secondly, increased contributions may also boost the credibility of the deposit guarantee system. A recent European initiative proposes further ambitious reforms. Besides a better consumer protection, the European deposit guarantee schemes would be largely harmonised, thus also promoting European financial integration. Risk-weighted financing of the schemes should counteract moral hazard, benefiting financial stability. However, this proposal has yet to be approved by the European Parliament and the Council. Its impact ought to be assessed in the light of the broader package of measures aimed at making the financial system more resilient, such as the new prudential supervision structure, the Basel III proposal for stricter capital and liquidity requirements, and the possible new levies on the financial sector.deposit assurance, financial institutions, financial crisis, moral hazard